( 465 ) 



XXI. — On the Determination of the Curve, on one of the coordinate planes, 

 which forms the Outer Limit of the Positions of the point of contact of an 

 Ellipsoid ivhich always touches the three planes of reference. By G- 

 Plarr, Docteur es-Sciences. Communicated by Professor Tait. 



(Read July 18, 1887.) 



INDEX TO CONTENTS. 



Section I. Notations 6, p 1 &c, p, cj>a>, 



,, II. Determines u 2 =Si4>~H, •y 2 =S,/(/>- l y, 



w 2 = S^^" 1 ^, 6 = iu + jv + kw, 



<p-H 

 Pi=— — > p = 9 + p x ; a^, a- , fluke. 



PAGE 



466 



u 



, 467 



III. 



IV 



VI 



VII 



VIII 



IX 



a=piq, li=pjq, y=pkq; p( )q opera- 

 tor of rotation. Differentiation of 

 o, P, y, <pw ; 80 = \p (, Sp 1 = if^e, 

 whence Sp = (ip + if^e. Expressions 

 of ip <ii, ifo'*") tyi" 1 ! ty\ t> > • • 468 



Various forms of the condition which 

 defines the limiting curve. Either 

 <|>e = 0, or Y.dp\f/e = 0, . . .470 



Both conditions give V. lp'fy'k = 0, hut 

 the second determines dp also, . 472 



Abandonment of the solution founded 

 on ^e = alone, .... 473 



Method founded on 4/ 'e = 0,\p (=0 ; 

 . • . <|»f= <hC) where £ = ix x +jy 1 + kz x , 475 



Remarkable expression of ty^; several 

 successive transformations relating 

 to it, 477 



Fundamental equation Y.dptye^O 

 giving TJ/'(Vidp) = 0, e being arbi- 

 trary. Treatment of ty'(Vidp) = Q 

 by S. Ogives dp parallel to \p^ what- 

 ever the direction of e, . . . 479 

 X. Treating v|/(Ci) = by S.t( ) and by 

 S.0 -1 £( ), d being a certain per- 

 pendicular to dp; the result in both 

 cases is 



W a 



u v 



= 0, 



PAGE 



V?h.' = a h bhChioT h = l, 2, 3, 



Section XL Treats $'& by S.<pi( ) = 0. There- 

 suit of the preceding section being 

 applied, the two equations obtained 



will contain - and — each equal 



v VJ 



to a rational function of Sai= -a lf 

 Sa/=-a 2 > & c '> & c -> * ne n i ne c °- 

 efficients of a, P, y, 

 ,, XII. Developments which show the presence 

 of the factor 



(a 2 -b i )(c 2 -a 2 )(b i -c 2 ) 

 a 2 b 2 c 2 



= -A 



482 



in all the terms of the equations. 

 Final form (I.) (II.) of the equa- 

 tion when the factor A is sup- 

 pressed, ..... 



XIII. Expressions for y and z, freed from 



radicals, except« = VS^ _1 i; other- 

 wise also: y expressed proportion- 

 ally to v, and z to w, . 



XIV. Expression of the nine coefficients 



Sai, So/, &c, in function of three 

 angles A, B, C, with their defini- 

 tion, ...... 



XV. Substitutions into (I.) and (II.), 

 XVI. Rationalising (I.) and (II.) in respect 



to - — . Two resulting equa- 

 te w 

 tions : the one of the second degree, 

 the other of the third degree in 

 tgB 



484 



487 



491 



493 

 494 



497 



The question is the following : — We consider the solid angle formed by 

 three planes at right angles to each other, and into the space of this single 

 octant we introduce a given ellipsoid, and cause its surface to be tangent to 



VOL. XXXIII. PART II. 3 Z 



