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



two arbitrary functions of the same variable may not simulta- 

 neously have very unequal forms under certain circumstances; 

 but when quite free and unrestricted, then they are identical. 



To enter into a further discussion of this fact of identicity 

 would lead me into details inconsistent with the object of the 

 present paper; but the utility and importance of it in the theory 

 of arbitrary functions are obvious to any one acquainted with 

 this calculus. 



Suppose we havefty x — i>ux, a. andy being given periodics 

 of the second order, and it be required to determine \J/ : 



Assume f^x = \J/a.r. <px v , which coincides with the ques- 

 tion when v — o whatever be the form of <p. Then substituting 

 olx for x, and taking they function on both sides, we get 



tyax=f {^x.<pax v } = ^~-, 

 tpx 



which differentiated with respect to v alone, and then reduced, 

 becomes /^ x logp«x _ <p*x . > 



yf. x . f '-px log p x Qx * 



putting^ for the differential function of f, and including in 

 the arbitrary function the log. 



A similar solution would come out, if we had as well as v 

 introduced an arbitrary factor b, and after the differentiation 

 with respect to b and v, put 6=1 and v=0. Both of these 

 methods, however, fail in giving the value of tyx when 



•** ■ is a constant quantity ; which happens in the simple 



cases of/\r being x, — x, or — . In these cases the solution is 



obtained by the extension of a neat and simple artifice em- 

 ployed by Mr. Babbage in the solution of fyx = 4>ct x. Sub- 

 stitute vi>ixx + b<px for ^/ux; change x into a.x; differentiate 

 with respect to v and b after eliminating \J/a.r; divide by dv 

 or db; and then putting u=l, 6=0, we obtain 



—ftyx=f'tyx.(tyx + fax) + <px. (5) 



Again : if we havef^i x = ■v{/« r x*, the condition being a."x=x 

 where n and r are any rational numbers whatever : 



Then assuming y\J/ x = \J/ ct r x . <px v . 

 we have by continually substituting for \J/ function its value 



4,x=f- l {($xff- l {(^x) v f- l {(?« 2r x ?.... 



where eachy* -1 applies to all the expression on the right 



* Since writing the above I have succeeded in discovering a solution to 

 this equation of the monomial form, even when r,n are irrational or ima- 

 ginary, hand 



