164 
MR. A. C, DIXON ON SIMULTANEOUS 
d (y,z), o, 
o, 
d ( x i>Pi)> (?i» q*}. 
(p 2 > ?i}. 
f/(.To,/q), 0, 
{Pa> ffa). 
d (y>P\)>Pi{q\>q 2 } 
PilPz, 2i) + P 2 {p 2 = 2 2 }> 
(10) d(z, p l ),q l \q l ,q 2 \, 
qAp^qA + qAv^qA, 
d (x u p a ), 0, 
(2i»Pi)» 
d ( x -2>P*)> {2],9a}, 
[q-2, pi], 
%>P2)>Pa{3i> ft). 
pAq^pA +Pa{?2»Pi} 
d { z >p 2 )’ q%{q\> q 2 }> 
2i(2nPi} + qdqt’Pi}, 
(15) d(x lt qj, {q. 2 , Pi), 
{pl’PA, 
d (x. 2 , q±), {q,,p,}, 
0, 
%n 2i)> pApppA + p 2 {q 2 , p 2 ], 
Pl{pl>pi}> 
d ( z >qi), qAq*> pA + q 2 {q 2 ,p 2 }, 
q\{p\>Pi}> 
c K x i> qAApi, qi K 
0, 
(20) d(x 2 , q 2 ),{p 2 , qA, 
fPi>P 2 }> 
d (y>qi)>Pi{Pi>qi} + Pa(p 2 , qA, 
P 2 (Pi>P 2 }> 
d ( z > q*), q\{p\> qA + 5 2 (p 2 , ?i}» 
2 2 (PuP 2 ) > 
d (pi,p*)> o, {xi,<h} + Pi{y,qA +qi{z,qi] + [ x ^qA -\-p. 2 {y,q2}+q 2 {z,q z }, 
d iPi> 2i)> { x \,qA +Pi{y, q 2 ) + ?i{*, 3a}, — {®i>Pa} -Pi{y,p. 2 ] - qiU,p 2 }> 
( 25 ) d(p liqz ), - {x^qj -jhiy^hs-qii z ,qi}> —{*2>Pt} —p*{y>Pi x > ~ qA z >?i>}> 
d (P2> qM x i> q-2 1 + pdy, q 2 ] + qd z >qz}> {^Pi} + pi{p>Pi) + qA^pA, 
— i x 2>qi} —p*{y>qi} — q2&qi}> i x ^pA +pAy,pA + qA z >pA> 
d (qi, qM x i» Pil +Pi{y> Pil + qA z >Pi) + {%,p 2 } +p 2 {y, p 3 } + q-A z > &}> °> 
(5) 
Here {^) l5 g,}, for instance, is written for d(fi,f 2 )/d(pi,qi), and every fifth row is 
numbered. 
§ 16. In order, then, to solve the equations f x — 0 , f 2 = 0 we have to form such a 
linear combination of the determinants of this array as will be a complete bidiffer¬ 
ential, say d(f<, f { ), f 2 , f 4 being such functions that the equations f x — 0 = f,, f 5 = ci 3 , 
f\ = <A t can be solved for yq, q l5 p 2 . q 2 . The array contains twenty-eight rows, but 
thirteen of these are combinations of the other fifteen. For instance, multiply the first 
row by df/dx 2 , the second by dfjdy, the third by df^/dz, the seventh by cfjdp^ the 
eleventh by ofJdp. 2> the fifteenth by q/'iT' 2 i’ the nineteenth by cf l /oq. 2 and add ; the 
resulting row is 
d(x u f } ), 0 , 0 , 
which vanishes. Other vanishing rows may be formed similarly by combining the 
rows of the array so as to have in the first column one of the following— 
d ( x iJi), d (x. 2 ,f 1 ), %,/j), <%/), d(p l ,f 1 ), d (p. 2K fi), % 2 >/l)> 
d { x i,fA> d(x 2 ,fo), %,/,), d{z,f 2 ), d(p 1 ,f 2 ), ff(p 2 ,/ 2 ), d(q L ,f 2 ), d{q 2 ,f 2 ). 
