88 Prof. Challis on investigating generally the partial 



It will be convenient in the subsequent reasoning to adopt 

 some notation to designate when the differentiation of a quan- 

 tity is with respect to space, the time being given. I shall 

 continue to indicate a complete differential coefficient with 

 respect to time and space by putting it in brackets, but a dif- 

 ferential in brackets will show that the differentiation is with 



respect to space only. Thus ■ ^' is a fraction the numerator 



of which is the increment of ^ with respect to space, and the 

 denominator is the increment of the time. This being pre- 

 mised, from equation (G.) we obtain, by differentiating with 

 respect to time, 



and by differentiating this equation with respect to space, and 

 dividing by a given increment of the time, 



dt dt^ ' 



Therefore, 



_ -^dt _ iP_^ dn^ ci^ (IN (fj> 



dt ~ dt^ " di^ "^ dt^ 'dt ^ dt' dt^' 



Again, from (7.) a value of r— ^ > expressed in partial 



differential coefficients of N and <p, may be obtained. Equating 

 this to the value above, and eliminating N by means of (4.), the 

 result is the required equation in partial differential coefl[i- 

 cients of $. This equation is of the third order, and is very 

 complicated, but, I believe, admits of much simplification. My 

 present limits do not allow of pursuing this inquiry further, 

 and I am desirous of making one more deduction from equa- 

 tion (2.) 



It is shown above that -y^ + ^-r^ = 0. But with refer- 

 dt dt 



("•§) 



