184 



Professor Baker, On the stability of periodic 



of the form ilU, where JJ is periodic in t, having /x as a factor, 

 except the three /„ „ \ /™ ™ \ z™ ^ \ . 



any one of these, say {x^, Xg), is of the form {x,9, 2-,°) + ^ {Xr, x^), 

 where the notation { } must be observed. The matrix of four rows 

 and columns formed by the sixteen coefficients on the right, if we 

 retain only to the first power of fj,, may then be represented by 



, 0\ + {ji 



J \~{x, x}, - {x, y)) 



■ Ho, 



where the denotes a matrix of two rows and columns whose 

 elements are all zero, Hq denotes the value for Xj^ = x-^, x^ = x^, 

 of the matrix 



Fn 



8^0 

 dx-^ 



dxj_dx2 



Ho 



dx^ 



OdOsMjtJu-t 



{y, x) denotes for x^ = x^P, y^ = nj, + w^, the matrix 



{y, x) = 



dy-^x^ ' dy-^dx^ 

 d^F^ d^F^ 



dy.^x-^ ' dy^x^ 

 with similar meanings for {y, y) and {x, y), but {x, x) is such that 



dx-^^ \dx-^) 



(JJb-t wJun 



\dx-^xj ' 



_9^ 



\JtL/-t (JJUn 



djFo 



dx^ 



( ^'F, \ 

 Xdxjdx^J 



[dx^J 



f^ 



dxj^ 

 d^F^ 



dx^x^ 

 dx^^ 



d^Fr. 



f. 



t/zO ~~ <^0 3 GT' C.J Oj 11 Q 



wherein f -^ — ^j is the value of "^^ for Xj^ 



in the first matrix we are to substitute x^ = x,P + fiX^, and retain 

 only to the first power of /x, while in the second matrix we are to 

 put x^^, n^t + w,. for x^., yr- 



If the matrix of coefl&cients as so explained be denoted by 

 w + /iv we require, to solve these equations, to compute what, in 

 the notation of the paper referred to, is denoted by Q. {u + fiv), 

 which by a theorem there given (p. 159) is equal to 



a (u) Q. [{Q. (m))-vv a (u)]. 



u= / 0,0 



In case, as here, 



Ho, 



