integral 
traeted from another, the difference la tin- definite Integral 
from the I, rat viiliu 1 <>f tin- ii.ilfpt'mli-nt viiri.iMr I" tin- 
rtiviiml. 11 A is tin- iiidftlnito inteural of JJ relative' to (', 
tlirn:iN> \'.\- thr.lnirn-niial o.rHVinit of A relative toC. 
(6) An Integral of ft differential equation or system of such 
equal loan lit a system of a lower on I IT (it may be a single 
equation, and it may lie one or more ordinary equations) 
from which the Drat system U deduuible. If the onl' r "f 
the second system is lower than the tlrstby one, the former 
is :i //rv/ integral; if by two, u unco n't infi-i/fif, etc. A com- 
plete integral is one which contains the greatest possible 
1 1 u n it i. -i of arbitrary Constanta for an integral of that order. 
A singular intftjrtu is one which contains ;i .smalk-r niun- 
bur of arbitrary constants, but is not a particular case of 
any irreducible complete integral. A general inttyral is 
oni which contains the greatest possible number of arbi- 
trary functions; but the complete integral of an ordinary 
different ial equation U also tenued a general integral. A 
fHtrtit-nltir inti-'ir/if la a particular case of a complete Inte- 
gral having a smaller number of arbitrary constants, (c) 
A quantity or expression which a system of differential 
equations mokea to be constant is also termed an integral 
of that system. Abelian Integral. See Abelian. Cir- 
cular integral, an Integral taken round a circle in the 
plane of the imaginary variable, any pole, of the function 
being the center. Closed integral, ;in imaginary Inte- 
gral whose upper and lower limits coincide, a circuit being 
described by tne variable in the course of the integration. 
Complete Integral ee complete. Cosine integral. 
See cwnne. Dirtcnletian integral* an integral of the 
form 
which for A = oo has a finite and determinate value other 
than zero and independent of a. Such, for example, Is 
r a sin hx . 
Elliptic, Eulerian, exponential Integral. See the ad- 
Jectfves. First Integral, the result of performing the 
operation of taking the integral once. Fourierian inte- 
gral, a double integral of the form 
A/" 
J o J o 
which, after the performance of the integration relatively 
to y, becomes a Uirlchletlan integral. Hyperelliptlc, 
Imaginary, etc.. Integral See the adjectives. Inte- 
gral Of the first kind, an Abelian integral lor which the 
second member of Abel a formula vanishes. Integral of 
the second kind, an Abulian integral for which the second 
member of Abel's formula is rational. Integral of the 
third kind, an Al>elian integral for which the second mem. 
her of Abel's formula involves a logarithmic function. 
Irreducible Integral, an integral not a rational integral 
homogeneous function of integrals of lower degree. Lin- 
ear Integral, an integral along one or more straight lines 
In the plane of the imaginary variable. Line-integral, 
surface-integral, volume-integral, having different 
values at different points of space, the integral obtained 
by breaking a curve, a surface, or a solid into equal ele- 
mentary portions, and taking the sum of the products ob- 
tained by multiplying each by the value of the quantity 
integrated at that point. Open integral, an integral 
whose two limits are unequal. 
integralism (iii'te-gral-izm), . [< integral + 
-i."n/.] Same as integrality. 
The philosophy developed from unlversology he [Ste- 
phen Pearl Andrews] called iiitegralitin. 
Appleton't A*n. Cyc., 1886, p. 663. 
integrality (iu-te-gral'i-ti), . [= P. integrattte; 
as integral 4- -i'ty.'] The quality of being in- 
tegral; entireness. [Bare.] 
Such as iu their integrality support nature. 
Whitaker, Blood of the Grape. 
integrally (in'te-gral-i), adv. In an integral 
manner; wholly; completely. 
integrant (in'te-grant), a. and n. [= F. \ntt- 
grant= Sp. Pg. It. integrantc, < L. integran(t-)s, 
ppr. of inteyrarc, make whole, repair, renew: 
see integrate.'] I. a. 1. Going to the formation 
of an integral whole. 
In the integrate whole of a human body, the head, body, 
and limbs, its integrant parts, are not contained in, but 
each lies out of, each other. Hamilton. 
If the sun was not created till the Fourth Day, what be- 
comes of the astronomic teaching that earth has been from 
the beginning an integrant part of the solar system? 
O. D. Boardman, Creative Week, p. 140. 
2. Intrinsic : same as integral, 3, but modified 
in form by an affectation of precision. 
A process ... of degeneration is an integrant and ac- 
tive part of the economy of nature. 
Maudiley, Body and Will, p. 237. 
Integrant molecule, in Haiiy's theory of crystals, the 
smallest particle of a crystal that can be arrived at by me- 
chanical division. 
II, n. All integral part. 
integrate (in'to-grai), v. t.\ pret. and pp. int<- 
gnitrtl. ppr. (nhgroting. [< Ij. inli i/riiti<g, pp. 
of integrarr (> It. intt/inire = Sp. Pg. intfijnn- 
= F. integrer), make whole, renew, repair, be- 
gin again, < intr<i<r, whole, frosli: see iii/<ii'-r.~\ 
1. To bring together the parts of; bring to- 
gether as parts; segregate aud bring together 
like particle-. 
All the world must grant that two distinct substances, 
the soul and tho body, go to compound ami inl<-imif'- the 
SoMh, Work., VIL 1. 
3131 
There is a property in the horizon which no man has 
but he whose eye can integrate all the parU that Is, the 
poet. Etnenon, Nature. 
2. To perform the mathematical operation of 
integration. The mean value of a quantity over a space 
or time U obtained by Integrating that quantity ; hence, 
instruments which register the mean values of quantities 
or the totals of their instantaneous effects are called in- 
tegrating inntrutnenU : as, an integrating thermometer. 
Integrating factor. See factor. To Integrate a dif- 
ferential, in the integral calculus, to determine from that 
differential its primitive function, 
integrate (iu'te-grat), a. [< L. integratus, pp.: 
see the verb;] Summed up ; resulting from the 
aggregation of separate parts; complete. 
I'hi How liked you my quip to Hedou, about the gar- 
ter? Was 't not witty? 
Mor. Exceeding witty and integrate. 
B. Jonton, Cynthia's Bevels, II. 1. 
This whole is termed mathematical, because quantity is 
of mathematical consideration : vulgarly, Integral, more 
properly, inttyrate. lluryemdiciui, tr. by a Gentleman. 
integration (in-te-gra'shpn). n. [= F. inte- 
gration = Sp. integration = Pg. iutegraySo = It. 
integrazione. < LL. integratio(n-), a renewing, 
restoring, < L. intcgrare, renew, restore : see 
integrate.} 1 . The act of integrating, or bring- 
ing togeth'er the parts of an integral whole; 
the act of segregating and bringing together 
similar particles. 
Integration of parts means the connected play of them, 
so that, one being affected, the rest are affected. 
W. K. Clifford, Lectures, I. 95. 
The term integration we have already partly defined as 
the concentration of the material units which go to make 
up any aggregate. But a complete definition must rec- 
ognize the fact that along with the integration of wholes, 
there goes on (in all cases In which structural complexity 
is attained) an integration of parts. This secondary inte- 
gration may be defined as the segregation, or grouping to- 
gether, of those units of a heterogeneous aggregate which 
resemble one another. A good example fs afforded by 
crystallization. . . . Integration is seen in the rising of 
cream upon the surface of a dish of milk, and In the frothy 
collection of carbonic acid bubbles covering a newly. tilled 
glass of ale. J. Fitte, Cosmic 1'hilos., I. 336, 
2. In math., the operation inverse to differ- 
entiation; the operation of finding the integral 
of a function or of an equation. 3. The infer- 
ence of subcontrariety from " Some A is B " 
to " Some A is not B." Constant of integration, 
the constant which must be added to every integral with 
one limit fixed, in order to get the complete expression 
for an indefinite Integral : denoted by the letter C. Fi- 
nite integration, the summation of any number of terms 
of a series whoselaw is known. Gaussian method of 
approximate integration. See Gaiusian Indefi- 
nite, definite integration. See indefinite integral, un- 
der integral. Integration by parts, integration by the 
formula 
/Tivdt = u /"vdt - /frvdt) ~ . dt, 
by means of which many expressions are integrated. 
Integration by quadratures, the numerical approxlma- 
tion to the value of an Integral. Limits of integration, 
the initial and terminal values of the variable, oetween 
which a definite integral is taken. Path of Integration, 
the path on the plane of imaginary quantity along which 
a complex variable is supposed to vary In integration. 
Sign of integration, the character/, modified from a 
long 3 for suminn (sum), used to signify the process of in- 
tegration. It was Invented by Leibnitz, 
integrative (in'te-gra-tiv), a. [< integrate + 
4ve7] Tending to integrate or complete; con- 
ducive to integration or the formation of a 
whole. 
The integrative process which results in individual evo- 
lution. U. Spencer, Prin. of BloL, 8 383. 
integrator (in'te-gra-tor), n. [X integrate + 
-or.} An instrument for performing numerical 
integrations. There are a great variety of such instru- 
ments, as planimeters, tide-integrating machines, inte- 
grating thermometers, etc. 
integripalliate (in'te-gri-pal'i-at), a. An in- 
frequent but more correct form of integropal- 
liate. 
integrity (in-teg'ri-ti), . [= P. integriti = Pr. 
integritat = Sp. integridad = Pg. integridade 
= It. integrity, < L. integrita(t-)s, unimpaired 
condition, wholeness, entireness, purity, inno- 
cence, honesty, < integer, untouched, unimpair- 
ed, whole: see integer. From L. integrita(t-)x, 
through the OF., comes E. entirety, q. v.] 1. 
The state of being integral ; unimpaired extent, 
amount, or constitution ; wholeness ; complete- 
ness. 
In Japanese eyes every alien became a Bateren (padre), 
and therefore an evil person harbouring mischievous de- 
signs against the intciritii of the empire. 
Fortnightly Ker., S. S., XLI. 681. 
Left Valve of Oyster (Oitr 
iHiana), showing unbroken 
To violate the integrity of one part of the Key of India 
Is to impair the value of the whole of it 
Marvin, Gates of Herat, v. 
2. Unimpaired condition ; soundness of state ; 
freedom from corruption or impurity. 
intellect 
Your dishonour 
Mangles true Judgment, and bereaves the state 
i )f that integrity which should become it. 
Shot., Cor., 111. 1, l&s. 
We plead for no more but that the Church of God may 
have the same purity and integrity which It had In tin- 
primitive times. SttUingfttet, Sermons, 1. ix. 
3. Unimpaired morality; soundness of moral 
principle and character; entire uprightness or 
fidelity. 
The moral grandeur of independent integrity is the 
subllmest thing in nature. sV MWd ' 
Our foe. 
Tempting, affronts us with his foul esteem 
Of our integrity. Milton, i 1 . L., Ix. 320. 
There Is no surer mark of integrity than a courageous 
adherence to virtue in the midst of a general and scan- 
dalous apostacy. /,'/. Atterbury, Sermons, IL xli. 
Law of integrity, in logic, the principle that in any in- 
quiry all the known facts should be taken into account. 
= Syn, 1. Completeness. 3. Probity, Uyriglitnett, etc. 
See hmiesty. 
Integropallia (in'tf-gro-pal'i-a), n. pi. [NL., 
pi. of * tntegropaUix : see iategri>]>alluil.'] A sub- 
division of the lamellibrauchiate mollusks, in 
which the pallial line in the interior of tho 
shell is unbroken in its curvature and presents 
no indentation, and which have either no si- 
phons or short unretractile ones. 
integropallial (in'te-gro-paPi-al>, a. [< L. inte- 
ger, whole, + pallium, mantle.] Same as M- 
tegrojialliatf. 
integropalliate (in'te-gro-pal'i-at), a. [< L. 
integer, whole, 4- pallium, mantle: see palliate.'] 
In conch., having 
the pallial line in- 
tegral or unbroken 
by a notch or sinus, 
as a bivalve mollusk 
or lamellibranch : 
opposed to sinupal- 
liate. Also integri- 
palliate, integropal- 
lial. 
InttgropallUiic and si- 
nnpalliate, . . . applied 
to Lamellibranchs which have the pallial linecvenly round- 
ed or notched. Uuxiry, Anat. Invert, p. 413. 
integument (in-teg'u-ment), w. [= F. integu- 
ment = Sp. It. iiiteguinento, < L. integtimentum, 
a covering, < integere, cover, < in, upon, + te- 
gere, cover: see tegmen, tegument.'] 1. In gen- 
eral, a covering; that which covers or clothes. 
Many and much In price 
Were those integument* they wrought, t' adornc thy ex- 
equies. I'luii limn Iliad, xxil. 
Specifically 2. That which naturally covers 
or invests any animal or vegetable body, as a 
skin, shell, case, crust, or rind; especially, a 
continuous investment or covering, as the cu- 
taneous envelop or skin of an animal body, 
with or without its special appendages. The 
Integument may be thin, soft, and memnranous, as a flexi- 
ble skin, or variously thickened, hardened, crustaceous, 
chitinoiis, etc., as the shells of crustaceans and mollusks 
or the hard cases of insects ; and it often bears particular 
outgrowths or appendages, as hairs, feathers, or scales, 
integumental (in-tg-u-meu'tal), a. [< integu- 
ment + -/.] Same as integumentary. 
An inteyumental pit or genital cloaca. 
Iluxli'i/ and Martin, Elementary Biology, p. 276. 
integumentary (iu-teg-u-men'ta-ri), a. < in- 
tegument + -an/.] 1. (Covering or investing in 
general, as a skin, rind, or peel. 2. Pertain- 
ing to or consisting of integument ; tegumen- 
tary; integumental; cutaneous. 
integumentation (in-teg'u-men-ta'shon), . 
[< integument + -ation."\ The act of covering 
with integument; the covering itself. 
intellect (in'te-lekt), n. [= F. intellect = Sp. 
(obs.) intelecto = Pg. iiitellecto = It. intrlletto, < 
L. intellectux, a perceiving, perception, under- 
standing, < intetlegere, intelligere, perceive, un- 
derstand: see intelligent.'] 1. The understand- 
ing; the sum of all the cognitive faculties ex- 
cept sense, or except sense and imagination. 
The Latin word inteUectiu was used to translate the Greek 
row, which in the theory of Aristotle Is the faculty of the 
cognition of principles, and that which mainly distin- 
guishes man from the beasts. Hence, the psychologists of 
the Scotch school use intellect u the synonym of common 
entr, or the faculty of apprehending a priori principles. 
11\ts agent or actire intellect, according to Aristotle, is the 
impersonal Intellect that has created the world (see phrase 
below); the jxurirv, patient, or pottible intellect Is that 
which belongs to the individual and perishes with him. 
But with St Thomas Aquinas the distinction 1s quite dif- 
ferent, the jtnstible inteOfft being the faculty receptive of 
the Intelligible species emitted by things, while the agent 
intellect Is t lie power of operative thought The term mm 
intellect, said to be used by St Augustine, and certainly as 
early as Scotus Erigena, had always denoted the divine in- 
Mutt, unmixed with matter, until Kant (adopting, as was 
his f mjuent practice, the terminology of Luscher) applied 
it to intellect as separated, in its use or application, from 
