MATHEMATICAL NOTE*. 



Problem from the Papers of 1842. If F(x, y, z) = $ (w, v, to), 

 where F is homogeneous of the n ih degree in x, y, 0, and 



dF dF dF 



u= ^- t v= T~> w ~ ~T~ 

 dx ' _ dy dz 



Since 7^ is homogeneous of n dimensions in a?, ?/, z 9 we have 



dF 



nF= x -j- + y -j- + z -j- = xu + yv + zw. 

 dx 9 dy dz 



Hence ndF= xdu + ydv -f- zdw + wc?a; + vdy + wdz, 

 or (n 1) dF xdu + #cfo 



But (n- 



Therefore equating the coefficients of the differentials, 



.-(-i)g, ,-(.-i)g, .-(-1)2- 



* Cambridge Mathematical Journal, No. XV. Vol. m. p. 152, May, 1842. 



