480 Mr. W. H. L. Russell. [June 16, 



Next suppose that we transform by means of the equations — 



a\x, + \y + c Y z = e x , 



a 2 x + bzy + c 2 z = e 2 , 



and remember that 1? 2 , 3 being any functions — 



J<faj 0/(0?) 2 (t/) 3 z + j ^ 2 V 1 (» 3 (» 



+ jd« 3 'O) 0i 0) (p 2 (y) = 0i« 3 2/ 3 z; 



then, transforming as before (with similar limits), 

 + j <& 03^^ 0,fc^ 



= 01^1 02^2 02^2 — X 2 02/^2 03 ^2 > 



we observe there are three arbitrary functions, which can be taken at 

 pleasure, and six arbitrary constants. We perceive therefore that the 

 formula is very extensive. 



The limits are of course connected by the equations — 



cii\ + + c x v Y = e lt 

 a r\ + frp"'2 + c i I/ 2 = e i» 



a 2 \ x + Z> 2 / t l + C 2 I, l = e 2> 

 a 2^2 + + c 2 v g — H' 



Next let us transform by means of the equations — 



a^x 2 — 2mx) + \{^ — 2my) + c^—Zvz) = e v 

 a 2 (x^-2mx) + b 2 (y*-2my) + c^-lvz) = e 2 ; 

 then, proceeding as before, — 



