SIMILAR TO THOSE OF HYDRODYNAMICS. 



365 



In a paper by Professor H. TV. Lloyd Tanner "On the Transformation of a Linear 

 Differential Expression," contained in the Quarterly Journal of Mathematics for 

 December, 1878, it is proved that 



Uidxi + tt 2 dx 2 + 



+ Undx n 



is reducible to the form 



dvi + ic 2 dc 2 + Wtflv s + .... + io,dv r , 



if, and only if, 



(/3) 



= o, 



(11) 



where there are 2 r rows ; and that the same quantity is reducible to the form 



tCxdi'i + w 2 dv 2 +-.... + w,.dv r , 

 if, and only if, 



= 0, 



(12) 



where there are 2 r + 1 rows. In both of these cases the functions w and v, though 

 functions of x ly x 2 , .... x n , are not mutually dependent. In the case in hand it will be 

 shown that the expression (a) can always be reduced to the form (/3) when r = n + 1. 

 Tbe case when u^dx-^ + n 2 dx 2 + .... + i( n dx 2 is an exact differential d$ gives for <f>, 

 as already remarked, 



(13) 



d°-<f> d^ d*+ 



* + dx~? + + dxl = a 



dxi 



I will give here, for convenience of reference, the transformation of this equation in 

 polar coordinates. Some of the properties of the function <f> will be found in another 

 part of the paper. Denote by r, a,, a 2 , . . . . c^.j the polar coordinates of a point in 

 space of n dimensions. Then write 



vol. x. 47 



