NATURE 



[June 



1903 



It is stated (Art. 122) that there is a linear relation 

 between any three of the twenty-four integrals of the 

 equation. The limitation that it is essential to consider 

 only such groups of three integrals as have a common 

 domain does not appear until we reach Art. 124, where 

 it seems to contradict the statement in Art. 122. 



The twenty-four integrals are divided into six groups 

 of four each, and the members of each group of four are 

 described as being equal. It should be pointed out that 

 the members of each group of four fall into two pairs, 

 that the members of one of these pairs are equivalent to 

 one another, as they have the same domain ; but they do 

 not have the same domain as the members of the other 

 pair (which are equivalent to one another). The four 

 members of a group of four are equivalent to one another 

 only in the domain common to them all. The integrals 

 of one pair are to be regarded as continuations of the 

 integrals of the other pair. From this it follows that in 

 any lineu.i lelation between three of the integrals, it is 

 not possible to replace any integral by another member of 

 the group of four to which it belongs without examining 

 whether the integrals appearing in the ,final relation have 

 a common domain. 



For example, relation No. (vi.), p. 219, viz. : — 



Yi = M5Y5 + NgYe 

 is intelligible if we take 

 _ Yj = F(a, &, 7, x) 

 Yg = (I - x)"^ rfa, 7-/3, a - /3 4- I, _i_ "\ 



Ye = (I - ^) ~^ r(|^i3, 7 - «, /3 - o + I, -J-_ \ 



because these integrals have a common domain. But it 

 becomes meaningless if we replace 



Y5 by x"^ Y(a, 0-7-1- I,a-i3-|-i, }\ 



ngs to the same group of four inte 

 »usly taken for Y5 ; and if we replace 



Ygby ^"^Ffj3, /8 - 7 -t- I, 3 - a + I, i^ ; 



for Yi, Y5, Yg have now no common domain, except 

 possibly points on the unit circle. This peculiarity 

 had been noticed by Kummer in his memoir on the 

 hypergeometric series. He held that even supposing 

 we make the changes described above for Y5 and 

 Yg, the equation should not be rejected as meaningless ; 

 for the two sides are now the expansions of the same 

 function of ,r, one proceeding according to powers of ;r, 

 and convergent inside the unit circle, the other proceeding 



according to powers of — and convergent outside the 



unit circle ; and he illustrated the subject by deducing 

 from one side of one of the equations the expansion of 

 tan X in powers of .r, and from the other side of the 



equation its expansion in powers of — . 



The whole subject received a thorough revision by 



Goursat (in the Annales de I'Ecole Normale Sup^rieure, 



Ser. ii. t. x. 1881), who shows that in some cases the 



linear relations between the three integrals do not possess 



NO. 1754, VOL. 68] 



which belongs to the same group of four integrals as 

 that previously taken for Y5 ; and if we replace 



the same form throughout the whole of the plane of the 

 complex variable. There still remains, however, for 

 future researchers the discovery of an algebraic demon- 

 stration of such equations as the linear relation between 



F(a, &, 7, x), x^-y F(a-7+l, /8-7+I, 2-7, x), 

 and 



F(a, j8, a-f-/3-7-(-l, \ - x), 



series proceeding respectively according to integra 

 powers of .r, non-integral powers of r, and integra 

 powers of (i - x), where, however, the last series cannot 

 be expanded in integral powers of x. 



The following details may be noticed : — 



I. There is some obscurity in the explanation given in 

 the note to Art. 25.1 



If the system of curves /(;ir,/, c) — ohave a node-locus 

 let the node on the curve f{x,y, a) = o be given by 

 I = (a), r, = r/. (a). 



The node-locus will be found by eliminating a between 

 the last two equations. The point to be explained is the 

 reason for the appearance of this locus as a factor in the 

 equation Disct^ fkx->yi c) = o. 



The coordinates of the node on the curve f{x^ j, a-|-da 

 = o may be called | -I- 6|, »/ -f ^. Then, the 



following equations hold :—/(!, »?, a) =0, .L =0, -^ =0 ; 



O^ Or) 



and the equations which can be obtained from them by 

 changing |, »;, a into ^ -i- 8^, »; -H S^j « + S« respectively. 

 Of this last set of three only the first is required, viz. 

 /(^ -f 8^, >? -f ht], a -t- Sa) = o. Neglecting quantities of 

 the second order, and using the preceding equations, it 



Hence the values | = ^(a). 



follows that ~^ha =0. 

 oa 



Y] = Ma) satisfy ^ = o, as well as/= o, and therefore 

 Oa 



the node-locus is a factor of Disct^ f{x,y,c) = o. 



II. The properties of the Schwarzian derivative (Art. 62) 

 may be thrown into a more symmetric form, viz :— 



\s,x\ (dxf = - {x, s} (dsf 



{s,x\ (dxf ={s,y) {dyf + {y, xj {dxf 



III. In Art. 192, the argument may be stated thus : — 

 It is given that 



~aF aF 



raF aFi 

 Lar arjJ 



a [1,7,] -"• 



From this it follows that 



- r ,aF ap T. aF 1 



a [ f , r, ] -°- 



Hence the equation of the tangent plane to the surface 



i7f ^ • 9F , aF /.aF^ aF p\ 



.■^Y{x,y), VIZ. .= .g^- +^-a,-(^aT^Vr / 



can, by putting — = X, be expressed in the form 



a^ 



z = '\x +y(f>(^) + '>//■ (X), so that it is expressible in terms 

 of a single arbitrary parameter X. The quantities 



^. '?) ^) ^ are not all functions of a single parameter, 

 a^ dr] 



IV. The solutions of Laplace's equation, which have 



1 The word "discriminant-equation' in the fourth line should be 

 " diflferential equation.' 



