New Method of solving Equations of partial Differentials. 239 



takes place in Germany more frequently in the order of S. 

 through SW,, W., NW, N., NE., E., and SE., than in the 

 opposite order of S. through SE., E., NE., &c." 

 [To be continued.] 



XXVII. A new Method of solving Equations of partial Dif- 

 ferentials. By S. S. Greatheed, Esq., B.A., of Trinity 

 College, Cambridge.* 



SEPARATION of the symbols of operation from those of 

 quantity, has, as far as I know, been hitherto applied only 

 to the calculus of finite differences, and to the differential 

 calculus where both are involved. It appears to me that if 

 any much greater eminence than that to which analysis has 

 already been brought, remains to be attained by it, that 

 process is the most obvious and likely path. The following 

 pages will show how, by applying it, a large class of partial 

 differential equations, including nearly all that occur in ap- 

 plied mathematics, may be reduced to differential equations 

 of two variables. 



The following known theorem is one which will be con- 

 stantly made use of. 



The expression e dx f(x) is equivalent to Taylor's series 

 forf(x + h). 



I shall begin with the equation of the first order and degree 

 with constant coefficients : 



dz . dz 



a — + b -y- = c. 

 dx dy 



d z 

 If instead of -7— we had nz, n being a constant, the 



equation would become 



dz , , 

 a-r- + nbz — c, 

 dx 



a linear equation between x and 2, which may be solved by 



nbx 



multiplying by the integrating factor 1 « , whence 



d nbx c nix 



--—lea z) = — e a 

 dx\ ) a 



nbx nbx 



therefore * « * = J3 * ° + c > 



c - nb * 



z = —r + e a r - 



n o 



* Communicated by the Author. 



