I N C 



Cor. Having thus found the fum of the whole feries, we 

 may readily find the fum of as many terms as we pleafe, 

 for we have fecn that the fum beginning at the nth term from 



the fir ft, is ; -; and therefore, beginninj: at the 



2 n (n + i) 6 ° 



{n -t i)th term, the fum will be 



which lad fum taken from the whole fi 

 of the tirll it terms, that is, 



fum n terms = 



I N C 



2 (« + i) (« + 2)' 



um f , will leave that 



i!.Y = aJ 



rf Y = 



(/■ Y = 



d-Y = 

 Now tliefe values being fubflituted 

 ftall liave for the increment of Y, 

 AY=^ 



f- 2i.vd.v + 3.v*^> 



zbdK- + 6 .X dx 



6d«-' 



(a + 2 b> 

 Again, given Y = 



(« + 



:(n-r !)(.•! + 2) 4(« -f 



Thus, if n = I, fum = t ; if n = 2, fui 

 = 3, the fum = /„, &c. 

 Ex. 8. — Find the fum of the infinite feries 



11" + ^)- ' 



Here the general term of the feries being 



« (n + 4) 



increment of Y. 



d Y = ad> 



d^ Y = 

 d'Y = 

 fi'- Y = 



d'Y =^ 



Whence maki:i 

 we have 



3 a-) + A.V (i + 3.r) + Ax^ 

 + 6x' — c X' + .%■', to find the 



+ 2 i. »■</.! 



+ 2id X- 



— ^cx' dx + 4.1-' dx 



- 6,.vdx- + I2.x-dx- 

 — 6 c d x' + 24 J- (/.r' 



+ 24. dx* 



thefe fubftitutions in thi 



(hall, on the fame principles as in the precedmg examp! 

 have to find the integral of - 



AY 



~ 1+ A. 



general theorem. 



4)-v 



Now by art. 



/, r- = — X = ; becaufe A.r = — 4. 

 ^ (.v-4;*- -^•^- •'^ 4--^- 

 iviaking, therefore, .v = i , we have ^ for the fum of the 

 whole leries, beginning at the firll term. We might have 

 added here a v;uiaty of other examples, had the limits of 



Aa-(a + 23s:- 3<:a-^ + 4s:") + Ax"(* - 

 (r + 4 *•) + A .V. 

 Thefe examples will explain the method of applying the 

 theorem to any cafe that may arife, it being as univerfal in 

 its application in the theory of increments, or differences ; 

 as the binomial theorem is in tlie expanfion of roots and 

 powers ; but, like the latter, it was left by its author witl»- 

 out deraonllration : it has, however, received many fince from 

 feveral able mathematicians, though there is no one of the™ 

 perhaps thit is quite fo Hui factory as might be wiflied ; 

 Maclaurin's refts upon the fluxional calculus, which it would 

 bedefirable not to introduce, if it could be obtained from i 



the article admitted of it, but fuch ai are given wiU be found "''.f"^^^^^ "°J to mtroauce, it it couia be obtamed trom more 

 to apply in a number of cafes ; and will mdioate to the in- f;^ldent and obvious pnnc.ples. Another dem^onftrat.on of 

 telligent reade 



the method of application in many others, 

 which are reducible to fimilar principles. It will be proper, 

 however, befoi-e we difmii's this fubjecl entirely, to fay a few 

 words on the celebrated theorem of Dr. Taylor, the learned 

 author of tiiis theory. This theorem may be exprell'ed as 

 follows : 



Let Y'reprefentanyfunftion whateverof the variable quan- 

 tity X ; then if x be increafed by any difference ^ x, the 

 value Y, that is Y + A Y, becomes (adopting the ditl' 



tial notation) Y + A Y = Y -|- 

 A ^1 ^' Y A A^ ^' Y , 



dY 



' I.2.3rf.V= 



A x^ d' Y 



I.2.3.4-/.. 

 A x' d' Y 



A .V* d^ Y 



. , , — 1 T^ j\ T&c.wherethe 



law of continuation is obvious, and requires no farther de- 

 velopement ; it muft, however, be remarked, that when A x 

 is negative, the terms of the above feries mull be taken plus 

 and minus alternately ; and it will therefore be more ge- 

 nerally expreffed by 



A .v ^ Y ^ A^^d' Y ^ A x' d' Y 

 2"^.v' 



AY 



I dA 



■S'lx 



4- &c. 



In order then to find the increment of any function of a 

 variable quantity, we muft take the fuccelTivc order of its 

 fluxions, by whirh means all the fiuxional parts in botli tlie 

 numerator and denominator will difappear ; and we (hall 

 have the value of A Y expreffcd in terms of .r and A x; and 

 this exprelhon will always be finite, unlcfs the function be 

 tranfcendental. 



Let us propofe the funftion Y — ax + b x- -^ x' to 

 find the value gf Y', when * becomes x + S x. 



ims theorem, and which is confidered the moil fatisfadory 

 of any that has yet appeared, is given by I'Huilier of Ge- 

 neva, in his work entitled " Principiorum calculi diff. et 

 integr. expofitio elemenraris,'' in 4I0. publifhed m 1 795. 

 INCROACHMEN F, m Law. See Accroching. 

 INCRUSTATION, the lining or coating of a wall, 

 either with glolTy ftones, ruftics, marble, pottery, or ftuc- 

 co-work ; and that either equably, or in panels and com- 

 partiments. 



Incrustation, in Natural Hijiary, is one of the modes 

 in which organic remains are preferved in the earth : the 

 petrifying or incrulling matter, forming fo thin a coat or 

 crull around the extraneous foiTii, that its external form is 

 nearly preferved on the ftony incruilation. Mr. Willian» 

 Martin claffes thefe among the artificial genera of reliquia, 

 and obferves, " Outhnes," p. 177, note, that "the common 

 calcareous incru Rations of our rivers, &c. have been ranked 

 as petrifaftions ; but, with more propriety, may be con- 

 fidered as incipient matrices.'' If the coat of mineral matter 

 be too tliick to exhibit externally the exact figure of the 

 foflil which it covers, fuch are called by this author fub- 

 incrulling matrices^ of wliich irontlone nodules inclofing 

 leaves of plants. &c. are well-known examples. The petri- 

 fying fprings of Derbylhire, (for a lill of which, fee Mr. 

 Farcy's Report on that county, vol. i.) and other calcareous 

 dillricls, are wall known for the curious and whimfical in- 

 cruflations which are there in a (liort time produced. 



INCRUSTED, or iNCUusTATi-n Column. See Incruf. 

 tattd COLL.-UN. 



INCUBATION, in Comparative Anatomy, is the term 

 applied to the coi.dudl of molt birds in relling or fitting upon 

 their eggs, in order to communicate to them the iieceiT.iry 

 degree of warmth for the excitement of their internal parts, 

 and the developcment of tlie foetus. 



The phenomena produced by incubation are fo interefl. 

 C a .oit. 



