356 Dr. C. J. Hargreave on Differential 



each of which admits of an infinite variety of forms by reason of 

 each constant being arbitrary. If, therefore, we profess to make 

 the constants equal, we merely transfer the arbitrariness to the 

 functions, and obtain 



(o-/ I (.-©)(o-/ 2 (y-e"))(c-/ 3 (, + i))=0 



as the complete primitive. If we proceed further, and choose to 

 make the three functional forms identical, we have still made 

 no advance towards connecting the three solutions ; for the arbi- 

 trariness is merely transferred again from the functional symbols 

 to the respective arguments governed by them. These argu- 

 ments are not, as appears to be assumed, definite functions, but 

 each of them is just as arbitrary as if it were governed by an 

 arbitrary symbol. We have no more right to place the solu- 

 tion of 



in the form 



than in any other form such as 



= \og(y--x 3 J 3 





/ % 3 \ n / sc\ n ~ l , 



css v-~g) + a \y-v +•••' 



and so on indefinitely, the particular argument actually obtained 

 being the result of accident — that is, depending upon the parti- 

 cular integrating factor made use of in order to obtain it. It is 

 of course quite obvious that when the equation is once resolved 

 into n distinct equations, no connexion can exist between the 

 integrating factors which we may choose to employ for their 

 integration, since they are n totally distinct equations. 



This general method, therefore, is fallacious, if it professes to 

 do more than to integrate the n distinct equations into which 

 the differential equation of the nth. degree is divisible ; and if it 

 does not profess to do more than this, it is useless. 



On this subject, I refer to Professor Boole's excellent Treatise, 

 chap. vii. sect. 3. In that section, the complete primitive is 

 supposed to admit of the definite form 



( c _V,)(c-V 2 )..(o-V )! )=0. 



We may, however, substitute /iV, for V,, / 2 V 2 for V 2 , and so on 



