Dr. Booth on a Theorem in A?iatytic Geometry. 177 



" qu'on sert peut-etre plus encore la science en simplifiant, de 

 la sorte, des theories deja connues, qu'en l'enrichissant de 

 theories nouvelles, et c'est la un sujet auquel on ne saurait 

 s'appliquer avec trop de soin." — Annates de Mathematiques, 

 torn. xix. p. 338. 



Extending this remark to the simplification of the methods 

 of establishing theorems already known, and remarkable for 

 their difficulty, I am induced to give an exceedingly simple 

 demonstration of a theorem, which may be found at p. 342 of 

 Dupin's Developpements de Geometrie, where the accomplished 

 author bestows more than four quarto pages of analytical cal- 

 culation of extreme complexity on this theorem, and yet leaves 

 its solution incomplete. 



The following is the theorem to which I allude: — 



Three points assumed on a right line are always retained in 

 three fixed planes, any fourth point P in this right line will de- 

 scribe an ellipsoid, whose centre is the common intersection of 

 the three fixed planes. 



Let O x, O y, O z, be the 

 intersections of the three fixed 

 planes, Ox, O y, Oz being 

 the axes of coordinates, and 

 C P the moving right line in 

 any position, meeting the plane 

 of O x y in the assumed point 

 C ; let the distances of P to the 

 points in the planes of x y, y z, 

 z x be c, a,b; and let the an- 

 gles between the axes of x and 

 y,yz, andz x be v, A, jo.; through 

 P let three right lines be drawn 

 P m, Pn, P r, parallel to the 

 lines O x, Oy, O * ; in the line 

 P C assume the point Q, so that P Q = 8, and complete the 

 parallelopiped of which P Q is the diagonal ; let the sides of 

 this parallelopiped parallel to the axes O x, O y, O z be «, |3, y, 

 then we shall have by a well-known theorem, given in most 

 elementary works on the subject*, which expresses the relation 

 between the diagonal sides and contained angles of a paralle- 

 lopiped. 



a 2 + ^ + y 2 + 2 /3 y cos A + 2 a y cos p. + 2 a |3 cos v 

 or dividing by 8 9 , 



8»! 



g2 + 



P . r 9 , *lrL-\ , « a y — .. . ««£ 



+ -&+ 2^ cos A + 2-f cosju, + 2-^cosv=l (1.) 



* See Legendre's Geometry, p. 249 (Brewster's Edition). 

 Phil. Mag. S. 3. Vol. 21. No. 137. Sept. 1842. N 



