336 Mr. J. Herapath on the Transformation of the Solutions 



it should therefore invariably maintain, renders the proposed 

 equation impossible in a general point of view. It should not 

 however hence be inferred, that when an equation is generally 

 it is universally impossible. Restrictions may be introduced 

 which may render a generally impossible equation, particularly 

 possible ; as for instance, in the problem of tangents consk 

 dered by Messrs. Euler, Wallace, Ivory, Herschel, &c. 



From either of the expressions (7) or (10) the form of tyx 

 may. be determined, and hence the general transformation of 

 the complete solution (1) into any other complete solution, 

 where the form without the arbitrary function coincides with 

 any particular form we please. For suppose a n x = x, and 

 that our complete solution is <p* r <p~ l x, it is required to find 

 the form of <J>, so that this complete solution may be trans- 

 formed into a complete solution with the particular form 



J* x. We have then f ( x = <p<x r f~ l x, or f t <px=$ct r x to find 

 the form of f involving an arbitrary function. The condition 



tn 



of f in this case is evidently f r x — x. By substituting in 

 (7) or (10) f* forf, and <f>for vf/, we have an equation involv- 

 ing only fx and arbitrary functions of x, a. x, a. r x, &c, from 

 which fx may be determined in a given function of an arbi- 

 trary function of x, and the various orders of « r x. Giving 

 to the arbitrary function a particular form, we shall have 

 j3 x for some particular form of $, and therefore generally 



$x = $fix and <p~ x = /3 $ x 

 where f y may be perfectly arbitrary. Consequently 



/t r _ — 1 _ n r „ — 1 —1 



x = 4>« <p x—^fia. $ <p y x, 



and the complete solution transformed into the particular form 



fsia ^ yx^fj'f-^-, 



in which \J/ r x = x. 



The brevity to which I am confined in such a paper as the 

 present, prevents me from detailing the numerous applications 

 and powers of the two important theorems we have just de- 

 duced. I shall therefore content myself with showing their 

 utility in the solution of a problem not, I believe, heretofore 

 attempted. 



Let us have given the form off r x, the condition being 



/ 



