Zz 



certain Differential Equations. 371 



Ex. 4. rx'^ + 2sxy + ty'^+{a%ci—n.n-if\)z=0. 

 Its solution from (2) is evidently 



'='*""(^"'"0T''''|^"'''^"2 + '^2|sin« -Z^]. (19) 



Ex. 5. ?-«^ + 2sxy + tif + {a%-2—n .n + l)s=0. 

 The solution is, by (9), 



2,_^»+i^_^-ij |^n^,|cos«'v/Ml^+-^2^sina v'mIIJ. (20) 



^a;. 6. 



The solution of this equation depends on that of (8), consequently 

 we have 



''=id^''"y[^l-'^^^''^^2 + ^/-sma\/^,'j. (21) 



Ex 7 fn \ 



rx^+'i'Sxy^-ty'^^-y--\.\Ypx->rqy) + a%_riS—^. 



The equation evidently depends for its solution on the differen- 

 tial equation / fn \ \ 



the solution of which, by (12), is 



. (2a _» \ 



y=A.cos( — X 2 +a). 



Accordingly the proposed partial differential equation has for its 

 solution 



, y 2a , — , y . 2a , 



^ = l/ri|cOS-A/M_„+./r2^Sm-^W_„. . . (22) 



Ex.%. , 2(m + 1) fl2 



X ^ x^ 

 This equation is obtained from (8) by substituting ^/~—ia — 



for « in the corresponding differential equation, consequently its 

 solutionis / j \nr / „\ / „\-\ 



It is unnecessary to add any further examples of this method, 

 as the foregoing are sufficient to show its appUcatiou. Most of 

 the partial differential equations hitherto treated by the calculus 

 of operations are readily seen to be simple cases of this method 

 of reduction by transformation. 



Trinity College, Dublin, 

 March 5, 185C. 



