250 Sir James Cockle on 



of the respective forms 



wherein a 2? &), and the m's are constants. 



72. In general we should have 



q = mi t- 2 + m 2 tT- 1 +n 2 t*T- 2 , 



r = m 3 t- 3 + mj!- 1 + n 3 t d T~ 2 + rc/T" 3 ; 



but when (21) is binomial the w's vanish. 



73. Now if to and A be determined from 



(» + 2) ((B+ 3) = J; A=| {«, + !+ 1=^}, 



all the n's will vanish if 



„ A cot 2 



wherein a 2 is arbitrary. 



74. Let 0= — 2 (ft) + 3) ; then p is obtained from X by the 

 substitution of a 2 for A, and of 6 for g>. 



75. Putting — ±b=Y, and recurring to arts. 18 and 32, it 

 will be seen that it is in fact the system (18) of art. 64 which 

 is now under discussion. 



76. Introducing a quantity K, defined by 



K 



-^-iA 

 ~4 2 V9 /' 



we get 



™ 2 =^V 2 + 2ft)(A + l)+4(ft) + 3)(a 2 + l), 



m 1 =a 2 {a 2 — 1) — A(A— 1) + g K, 



3 

 4 2 



m 3 = a 2 3 — 3a 2 2 + 2a 2 + K [a 2 — 2) 



+ A(2A 2 -3a 2 A + 3a 2 -f 2K-2), 



3 2 3 2 



m 4 =2^V 2 + ^-Y 2 A — 2K&) + 3a 2 m 2 — 3^m x 



+ 6a 2 2 ^-6A 2 ft) + 6(ft) + 2). 

 77. Multiplying the binomial terordinal into Tt d , substitu- 

 ting for T and putting D = t -j, we get 



(l + ^ 3 )D(D-l)(D-2)r+3{a 2 +(a 2 -6>)^}D(D-l)? 



+ 3 { m x + {m x + m 2 ) t d } D J + { m 3 + (m 3 + mj * 3 } £= 0. 



