1893.] of circles; and other systems of two tetrads. 57 



quadric surface two tangent planes each touching the three cones 

 which have their vertices in the points d^^, f/gj , c^j2 respectively ; 

 and the section by either tangent plane is thus a section touching 

 each of the three given sections 1, 2, 3 ; we have thus the eight 

 tangent sections of these three sections. 



7. Taking as three of the red sections the arbitrary sections 

 1, 2, 3 ; and grouping together two at pleasure of these sections, 

 say 2 and 3 ; we may take for the blue sections the two sections 

 through the axis d^^d^^d^^, and those through the axis d^^i^^i^^; we 

 have thus the four blue sections touching each of the given red 

 sections 1, 2, 3 ; and this being so, there exists a remaining red 

 section 4 touching each of the blue sections ; we have thus the 

 four blue sections touching each of the red sections 1, 2, 3 and 4. 

 This implies that the vertices or points d^^ and d^ lie on the axis 

 ^23^81^12' ^^^ thoA, the vertices or points ^^^ and \^ lie on the axis 

 ^23\ihi'i *^^' what is the same thing that the four sections 1, 2, 3, 4 

 have in conunon an axis ^^23*^21^31^24^34 ^^^ ^^^° ^^^ ^-^^^ '^23'^'2i*3iV34* 



8. If the quadric surface be a flat surface {surface aplatie) or 

 conic, then the red sections become chords of the conic ; the axes 

 are lines in the plane of the conic, and thus the tangent planes 

 through an axis each coincide with the plane of the conic, and 

 it would at first sight appear that any theorem as to tangency 

 becomes nugatory. But this is not so ; comparing with the last 

 preceding paragraph, we still have the theorem : on a given conic, 

 taking at pleasure any three chords 1, 2, 3, it is possible to find a 

 fourth chord 4, such that the four chords have in common an axis 



^23<^2i*^3i^24^^34 ^^^"^ ^^^^ ^"^ ^^^^ ^iihihihih^- ^^^"^ ^^® aualytlcal 

 theory (although somewhat complex) is extremely interesting. 

 Considering the conic xz — y^ = 0, the coordinates of a point on 

 the conic are given by x\y:z = \ :d -.6^, or say any point of the 

 conic is determined by its parameter 6 ; and this being so, consider- 

 ing any three chords 1, 2, 3, I take as for the two extremities of 1 

 the values e, ^ ; for those of 2 the values a, /S ; and for those of 3 

 the values 7, 3 ; the remaining chord 4 is to be determined as 

 above, and I take for its two extremities the values E, Z. 



9. Starting with the chords 1, 2, 3, we have each of the 

 points c?23' <^c. as the intersection of two lines, viz. these are 



Ja-aS -y {cL-\-h) + z = 0, ^. ja;a7 - y (a + y) + z = 0, 



w^y-y{^+j)+z^O, ''\x^8-y{(3 + S)+z = 0, 



xa^-y (a+ ^) + z ^0, . (xae - y (a + e) + ^ = 0, 



a:^e-y{^+e)+z^O, " '^^ [w^^- y (^ + + ^ = 0, 



^7^-2/(7+0+^=0, . ^xy€ -y (j + e) +z = 0, 



xSe-y{8+e) + z = 0, " ^'' [wS^ - y {8 + ^ i- z = 0, 



VOL. VIII. PT. II. 5 



