FUNCTION 



periodic occurrence, are the great procetwe* of phenomena may be thus arranged in diagrammatic 

 growth and reproduction. Or the various vital I fashion : 



Assimilation. 



Reception of Food 



Sennory and Nervoim. 

 iiHcular or Contractile. 

 Glandular or Secretory. 



r 



v ; 



Storage of 

 reserve 

 products. 



'* y Excretory. 

 3 Storage of 



Y products. 



Exit of waste 



products, 



Respiration. heat, &c. 



Income. Expenditure. 



Growth. Reproduction. 



In a singh-celled organism, such as an Amreba, 

 all the vital processes take place within narrow 

 limits, and just because of the simplicity of struc- 

 ture there must be great complexity of function 

 Compared with what occurs in a single cell of one 

 of the higher organisms". For here division of 

 labour is possible, and in the different cells special 

 functions predominate over the others. Thus, a 

 muscle-cell is contractile but not strictly nervous, 

 and a glandular cell is secretory without being 

 definitely contractile. With the division of labour 

 and resultant complexity of structure in a higher 

 organism, various functions appear which are only 

 foreshadowed in a protozoon. buch, for instance, is 

 the circulatory function, establishing nutritive and 

 respiratory communication between the distant 

 parts. But such a multiple process can readily be 

 seen to be the sum of several more fundamental 

 functions. It must also be noted that, while a 

 cell, tissue, or organ may have one dominant func- 

 tion, it may at the same time retain several sub- 

 functions. 



Another fact of general importance is the change 

 of function which may be exhibited by the same 

 organ in the course of its history that is to say, 

 through an ascending series of animals, or even in 

 the development of an individual. Thus, what is 

 a mere bladder, of little apparent account, near the 

 him! end of a frog's gut, becomes the respiratory 

 and sometimes nutritive Allantois (q.v.) or reptile 

 and bird, and an important part of the Placenta 

 {q.v.) in placental mammals. The importance of 

 this in relation to the general theory of evolution 

 has been emphasised by Dohrn in what he terms 

 the principle of functional change. 



Fundamentally, the functions of organs, the 

 properties of tissues, -the activities of cells, are 

 reducible to chemical changes in the living matter 

 or protoplasm. To the constant change in the 

 protoplasm the general term ' metabolism ' is ap- 

 plied, while this is again subdivided into processes 

 of upbuilding, construction, chemical synthesis, or 

 'anabolism,' and reverse processes of down-break- 

 ing, chemical disruption, or ' katabolism.' See 

 AMOEBA, BIOLOGY, CELL, PHYSIOLOGY, PROTO- 

 H.\SM, and the various functions, DIGESTION, 

 &c. In speaking of disease, 'functional' is opposed 

 to ' organic.' 



Function. When two quantities are so related 

 that a change in the one produces a corresponding 

 change in the other, the latter is termed a function 

 of the former. For example, the area of a triangle 

 is a function of the base, since the area decreases or 

 increases with the decrease or increase of the Iwise, 

 the altitude remaining unchanged. Again, if u = 

 ax* + bx + c, where a, b, and c are constant 

 quantities, and u and x variables ; then " is said to 

 be a function of x, since, by assigning to x a series 

 211 



of different values, a corresponding series of values 

 of u is obtained, showing its dependence on the 

 value given to x. Moreover, for this reason, x is 

 termed the independent, u the dependent variable. 

 There may be more than one independent variable 

 e.g. the area of a triangle depenus on its altitude 

 and its base, and is thus a function of two vari- 

 ables. Functionality, in algebra, is denoted by the 

 letters F, f, <j>, *, &c. Thus, that u is a function 

 of x may be denoted by the equation u = F(x) ; or, 

 if the value of u depends on more than one variable, 

 say upon x, y, and z, then by u = F(x, y, z). 



Functions are primarily classified as algebraical 

 or transcendental. The former include only those 

 functions which may be expressed in a finite 

 number of terms, involving only the elementary 

 algebraical operations of addition, subtraction, 

 multiplication, division, and root extraction. Several 

 terms are employed to denote the particular nature 

 of such functions. A rational function is one in 

 which there are no fractional powers of the variable 

 or variables ; integral functions do not include the 

 operation of division in any of their terms ; a homo- 

 geneous function is one in which the terms are all of 

 the same degree i.e. the sum of the indices of the 

 variables in each term is the same for every term. 

 For example, 



x 4 + x?y + y?y* + xy* + y* 



is a rational, integral, homogeneous function of the 

 fourth degree in x and y. Transcendental functions 

 are those which cannot l>e expressed in a finite 

 number of terms ; the principal types are ( 1 ) the 

 exponential function e*, and its inverse, log x; 

 (2) the circular functions, such as sin x, cos x, 

 tan x, &c., and their respective inverses, sin *'z, 

 cos ~ l x, tan ~*x, &c. 



Functions are also distinguished as continuous or 

 discontinuous. Any function is said to be continu- 

 ous when an infinitely small change in the value of 

 the independent variable produces only an infinitely 

 small change in the dependent variable ; and to be 

 discontinuous when an infinitely small change in 

 the independent variable makes a change in the 

 dependent variable either finite or infinitely great. 

 All purely algebraic expressions are continuous 

 functions ; as are also sucn transcendental functions 

 as (*, log x, sin x, cos x. 



Harmonic or periodic functions are those whose 

 values fluctuate regularly letween certain assigned 

 limits, passing through all their possible values, 

 while the independent variable changes by a certain 

 amount known as the period. Such functions are 

 of great importance in the theory of sound, as well 

 as in many other branches of mathematical physics. 

 Their essential feature is that, if f(x) be a periodic 

 function whose period is a, tnen /(* + $a) = 

 f(x - Ja), for all values of x. 



