720 MR J. H. MACLAGAN-WEDDERBURN ON 



19. Reduction to normal form. — If <p is not degenerate, we may, without loss of 

 generality, put <p = 1. The equation may then be written 



p + 4> l p + <f> 2 p = ct>a . . (22) 



If in (21) ts satisfies the equation 



we have 



ij +xv = '&~ 1 (f>a ■ . (23) 



where 



The reduction to this form may also be made by changing the independent variable. 

 Let ct be the new variable. Then we have 



p = ro± p = ^ d LP o + t^ d P . 

 drs cfcr 2 drs 



Hence on substituting in (22) we get 



CT 2 JP + (CT + <p,ti) d JL + <f>„p = <f>a 



which if 



nr= fdt¥(- fdfyj 

 reduces to 



20. The following method of solution is analogous to that given in paragraph 4 for 

 ordinary differential equations. 



It may be verified by direct calculation that the solution of 



p + ^p + ^p^O . . . (24) 



is 



p = (\+ fafbdt 2 + fafbfafbdt* . . . )a 



+ ( (cult + fa fb fadt 3 + ■ • . )/3 

 where a and /3 are arbitrary constant vectors and 



a = F( - /^eft) 6 = - G( ffadt)^ . 



21. If we put 



o- = ( fbdt + fb fa fbdfi . . . )a 

 + (l+ fbfadP . . . )/3 



