1872.] Mr. W. H. L. Russell on Linear Differential Equations. 19 



Hence, remembering the double sign of V y, 



vh' + yv 2 +pY=0, (1) 



a'+2 r ^ = 0, . , (2) 



r + yV + a> + 2 i u JT + ^ 2 + 2/3y jO 2 =0, (3) 



y'p+^-+3ppv+2 Pf iy=0. 



Trom the first two equations, 

 v - 



and, substituting in the fourth, 



v= -27 '° 2= 47 



h 2y 2 2y 



Consequently we shall have 



and the constants in the differential equation must satisfy the condition 

 derived from the third equation, 



2y 2 4y 2 * 



It is easy to see that a similar proof will apply in other cases. We will 

 now recapitulate our results. 



The solutions of differential equations, when they can be obtained in 

 finite terms, must be either algebraical, exponential, or logarithmic. If 

 algebraical, they must be either rational or irrational. The method of 

 finding the rational algebraical solutions has long been known. We have 

 endeavoured in the present paper to show how the irrational solutions may 

 be found for equations of the second order. We have proved that a 

 linear differential equation will not admit of a solution of the form f(e x ) ; 

 still less will it admit of a solution of the form/(e w ), when w is a function 

 of (#). The exponential solution must therefore be of the form Pe w , 

 where w is an invariant, that is, a function which remains unchanged 

 whatever algebraical value we may be able to assign to P. This invariant 

 enables us to reduce linear differential equations containing exponential 

 functions to other linear equations involving algebraical functions only. 

 We have, moreover, shown that logarithmic solutions of linear differen- 

 tial equations must in general be of the form Plog e Q, when P is an 

 algebraical function satisfying the proposed linear equation, and Q can 

 be deduced from another linear equation, which we have shown how to 

 construct. A great deal remains to be done ; nevertheless I think we see 

 not obscurely the true foundations on which the solution of linear dif- 

 ferential equations in finite terms must of necessity rest. 



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