EQUATION AND ITS TRANSFORMATIONS. 
811 
The form (37) is readily identified with (36), for, from (37), 
«=eW x ff cos 
\ct da J a 
™ 0 , i /1 d\^cos(f + «) .„ .. 
= Gr _ Lx^ +1 (- — j ---, if i=ax, 
= Caf +1 (- —Y cos ( ax + *) . 
\x dx) x 
A method of proving the theorems contained in Mr. Gaskin’s question is given 
in Hymers’s ‘ Treatise on Differentia] Equations, and on the Calculus of Finite 
Differences ’ (Cambridge, 1839) pp. 83-85. The result (38) is there verified by 
showing that 
(r 3 —a 3 ) /J cos ( xr-\-oC)clr 
satisfies the differential equation 
d 2 v 2p + 2 dv 2 
ofi- T~ J r av —b ; 
dor x dx 
and it is remarked that (37) may be verified in a similar manner by showing that 
/ d \ P cos (x\/r + a.) 
V ~ \dr) ' 
r being put equal to a 3 , satisfies 
cPv 
dx 2 
2p dv 
x dx 
+ 
ci 2 v= 0/“ 
The integral (37) was subsequently obtained by Id. Leslie Ellis by a different 
process in the ‘ Cambridge Mathematical Journal,’! vol. ii., p. 195 (February, 1841). 
A full account of Ellis’s method, with its application to the equation in question, is 
given in De Morgan’s ‘ Differential and Integral Calculus/ pp. 701-703. 
In a paper, ‘ £ Demarques sur l’equation y"-\- y' -\-ny-0'” Liouville’s Journal, ’ 
vol. xi., 1S46, pp. 338-340), M. Lebesgue proved that the integrals of the equations 
A , 
dx 2_1 ~ 
2i d v i t\ 
* In the second edition (1858) of Hymers’s woi’k, only tlie proof that (38) satisfies the differential 
equation is given (p. 128), no reference being made to Mr. Gaskin’s other result. An account of Boole’s 
solution and method, taken from the ‘ Philosophical Transactions ’ for 1844, is however introduced on 
pp. 99-106. 
t “ On the Integration of Certain Differential Equations,” pp. 169-177, 193-201. 
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