AS COMMONLY INVESTIGATED. 237 



and as two factors P and Q might justly be regarded sls prime when they con- 

 tamed no common factor, we might define prime factors in this way; as, how- 

 ever, it is not always easy to find these common measures, we shall prefer a 

 definition that is equivalent, and say that factors P and Q of any given func- 



P 



tion F X, are prime when all the values of x that make P = 0, make 7=r vanish, 



P 



whilst all the values that make Q = make ^ infinite. 



The relative magnitude of functions that vanish when z = a must evidently 

 depend upon the nature of their factors; and it will, therefore, be convenient 

 to lay down as a definition that functions which are not prime 



12 3 X 



U, U, U, U . . . U, &c. 



are said to be arranged in the order of their magnitude when the fraction 



X4-1 



u . 



u 



formed by dividing any one of the functions by that which precedes it, is equal 

 to zero for the value of z that makes each of the factors vanish. 



Proceeding to the development of a function into a series S U, of functions 



more simple than itself, we observe that for any value a, of the independent 

 variable z, that renders the function finite and definite, we must have 



Fz = Fa + R; 

 where R vanishes for z = a. The nature and magnitude of R will depend al- 

 together on that of F z ; it may deviate either slowly or rapidly ftom the vanish- 

 ing value when z departs from the value a; and it may even become infinite 

 for an infinitely small alteration in z, or it may cease altogether to exist (as in 

 the theory of conjugate points)* when z takes any other value than this of the 

 sinffle existence zero. 



'&' 



* The imaginary values of an ordinate, when the function is illustrated by geometry, serve very 

 well to exhibit this distinction between y as absolutely non-existent and y as merely equal to zero, 

 a distinction on which the whole theory of conjugate points depends; but I have elsewhere re- 

 marked that when we regard the problem as one of pure numerical arrangement, and not as 

 illustrated by geometry, this advantage is lost, and we are left without a symbol to express a rela- 

 tion of so much importance. 



The reader wiU readily perceive that y = 0, and y = a ■«/ — 1, considered geometrically, 

 differ in their degree of non-existence; since adding b to both, in other words, shifting the origin 

 VII. 3 K 



