132 Mr. H. A. Webb and Dr. J. R. Airey on the 



I. ,y = M(«, 7, x) 

 satisfies the differential equation 



•3 + (7->£r*-0 (8) 



II. The complete solution o£ the differential equation (3) is- 

 y = A .M(«, 7, «)+.B .^"v. M(a— 7 + 1, 2-y, a?), (4), 



which we shall write for brevity 



y-M(«,7, «), (5) 



where A and B are arbitrary constants of integration ; except 

 only when <y is a positive integer, in which case* the coefficient 

 of B is either infinite or identical with the coefficient of A. 

 In this case the complete solution of (3) may be written 



y= [A+Clog#].M(a, 7, x) 



rwi 1, ,V «(»+!) V/l 1 11 -_Iy 

 + L 7 \a 7 < / 7(y + l)'l.2U « + l 7 7+1 2/ 



«(» + !)(« + 2) jg /l 1 J. 1 1_ 1 



7(7 + l)(7 + 2) *1.2.3\a + « + l + « + 2 7 7+I y + 2' 



2 3/ 

 + to infinity!, (6) 



where A and C are arbitrary constants of integration. 



III. M(ct,y,x) = e x .M(y-cc,y,-x) (7) 



tf l -vM( a -7+l, 2-7, x)=e x .x 1 -y.ll{l-* } 2- 7 , -x). (8) 



From (7) and (8) it follows that tables will not be required 

 for negative values of x, if the tables cover wide enough 

 ranges of a and 7. 



IV. The asymptotic expansion of M(a, 7, x) for large 

 values of x is 



* The situation is^ similar to that which, arises with Bessel's equation 

 whew n is a positive integer, and a new function is required for the second 

 solution. 



