( 471 ) 



XIII. — On the Determination of the Curve, on one of the Coordinate Planes, ivhich 

 forms the Outer Limit of the Positions of the Point of Contact of an Ellipsoid 

 of Revolution which always touches the Three Planes of Reference. By G. 

 Plare, Docteur es-Sciences. 



(Read 1st July 1889.) 



In a former paper {Trans. Roy. Soc. Edin., vol. xxxiii. part ii. p. 465) the ellipsoid 

 under consideration was supposed to have its three axes different from one another in 

 length. I consider now the more simpler case of an Ellipsoid of Revolution. The 

 same notations as in the former paper shall be made use of, and for the establishment of 

 some of the results which form the basis of the present paper, reference must be made to 

 the former. 



The condition (§ IV. of the former paper) 



i/,s = 

 leads to the consequence 



This is to be satisfied for all points on the limiting curve. 

 If we put generally for any point : 



K = Nyjs'j^'k = iM 1 + jM/ + &Mj", 



we get by the expressions (§ VI.) of \f/j , \p'k : 



Mj = R 1 R 2 -Q 2 

 M/ =P 1 Q -P 2 R 2 

 M/' = P 2 Q -PA- 



In the former paper we did not show the manner in which the coefficients PiP 2 , &c, 

 are obtained, we may therefore be allowed to show the method for one or two of them; 

 as, for example, 



P x = Si\Js 'j = Sjijsi . 

 By the expression § III. we have 



= --Si 2 d>-H-J-Sijd>- 1 j - -SiU-ik 



u r- v jy j w -r 



+^(v^->i + < r ^)+£± i siv-'i 



= -i^-i + W^' 



w " ' u 



Now 



V^- 1 i = Vjk^- 1 i = jSktp- 1 i — kSj^- 1 i . 



VOL. XXXV. PART II. (NO. 13). 4 I 



