EQUATION AND ITS TRANSFORMATIONS. 
827 
(xvii.) 1856. Williamson. “ On the Solution of Certain Differential Equations.'’ 
‘ Philosophical Magazine/ Fourth series, vol. xi., pp. 364-369. 
The general integral of the equation 
is given in the form 
d~u „ i(i + 1) 
—+aN,=. — j-u 
cix 4, x A 
u—Ax 
cos (ax + a ), 
and the solutions of Piccati’s and several other equations are also obtained. The 
symbolic expressions are developed by means of the theorem 
(Da- 1 )" = a“"D"- ^ + 1) a - ( , + i) D «-i q. ( ? DJM|±lXT±l) a -o+ 2) £ )»-2 _ 
±1.3 . . . (2n— l)a _(2/J - 1) (D —a -1 ), 
of which a proof is given. See art. 37. 
(xviii.) 1857. Donkin. “ On the Equation of Laplace’s Functions, &c.” ‘Philo¬ 
sophical Transactions’ for 1857, p. 44. 
The integral of the equation in (xvii.) is given in the form 
x’^D-j (<q sin ax + c 3 cos ax). 
This solution occurs in a note, as an example of the application of the general 
method of the paper to a particular equation. See art. 37. 
(xix.) 1871. Glaisher. “ On Piccati’s Equation.” ‘ Quarterly Journal of 
Mathematics,’ vol. xi. pp. 267-273. 
By means of the definite integral (31) of art. 36, the solution of Piccati’s equation 
is obtained in the forms 
u=z(z ( c i e * “hO 6 qZq )> & c - 
and the formulae (22) and (23) of art. 31 are proved. See arts. 31, 36. 
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