[187" 
188] 
129 
188. 
ON THE SIMULTANEOUS THANSFOEMATION OF TWO HOMO 
GENEOUS FUNCTIONS OF THE SECOND ORDER 
[From the Quarterly Mathematical Journal, vol n. (1858), pp. 192—195.] 
In a former paper with this title, Cambridge and Dublin Math. Journal, t. iv. 
[1849], pp. 47—50 [74], I gave (founded on the methods of Jacobi and Prof. Boole) 
a simple solution of the problem, but the solution may I think be presented in an 
improved form as follows, where as before I consider for greater convenience the case 
of three variables only. 
Suppose that by the linear transformation0 
(x, y, z) = ( a , /3 , y $3-, y 1} Zl ), 
I /3', 7' 
1 /3", i" 
we have identically 
(a, b, c, f, g, h ][x, y, zf = (a, , b, , c, , f lt g 1 , Jh y x , z x f, 
(A, B, C, F, G, HJfx, y, zf = (A lt B lt G u F lt G u y x , zff 
and write also 
(£i> Vi, £i) = ( a > a " $£ V> O- 
/3, /3', /3" 
7, 1 > l" 
1 I represent in this manner the system of equations 
and so in all like cases 
C. III. 
x = ax ± + /3yj + 7z i , &c. 
17