34^ 



NATURE 



[August 8, 1895 



few instances : I^t A) . . . A^fi be n + i points in a plane 

 space S//. Let I' Iw any other point. We then have one 

 linear relation 



a, A, + OjAj + . . . + <i„i.,A„+, -f pV = o. 



.Vssume outside the siMce any p<jim (^. Constnict the plane 

 sjxices QA, . . . Aa, <JAj . . . A«+,, . . . // + i in all, 

 anil cut them by some line joining the residual |xiint A/,-(-,, A, 

 . . . respectively with a point R on the line <^>l'. AVe thus 

 obtain /; + I new jxiints A'n+,, A\ .... which joined give a 

 plane space Zn, that cuts S/i always in one and the same plane 

 cut Sx-,, however we may choose O and R, which is related 

 to P and the configuration of the A in a |>eculiar manner. 

 To follow the difl'erent steiJS indicated, let us .assume 



pV = ,/Q + /R 



(the three (wints are collinear) ; therefore 



(i,.\, + a^\. + . . . + </Q + rR = o. 



Joining R with Aj, we obtain a line that contains the point 



J'J — , which as 



is also = - • - 



"; + 

 space Aj.\j . . 



Jasi so 



.7, + «.,+ .. . (- </ + r = O 



'^ "■ , lliat is contained in ihc plane 

 • +'/ 



A', is therefore = ' 



a, + /• 



. I _ rtcA., + rR 

 " ff., + R 



The line .\'| .\'.j contains the point 



(a, +r)A', -(a, + i-).V, . 



|7, - (Ij 



rJ|.\, - (JoA., 

 "1 ~ "i 



iliinear wiih .\,, .\j. The (ilane sp.ice S«-i contains 



that is 



therefore all the 



points thus formed, and the proposi- 



tion follows at once. 



Ill a similar way it may \m: proved that, if two (« -i- i ) |)yramids 

 in a .Sm are in i>ers|)ective, the intersection of coi responding sides,, 



in all, are .til contained in a Sn-,. We provf; this 



simply for « = 2, which is sufficient to exhibit the general nay 

 of proceeding. \x\ \ B C, .\' B' C Ije two triangles in |)er- 

 spective ; lei .\.\', B15', CC have |K)int I' in common. Then we 

 must have 



I' = <iA -frt'A' 



= AB + A'B' 



= <C + f'C 



Join .\B, .\'B'- Their intersection, from 



aA + a'A' = *B + A'B' 

 follows 



aA - <iB_a'A'-i»'B' 

 a - b a' - «'■ ■ 



aA - m 6K - .C «C - aA 

 ■ /' - <■ ' 



Two plane S|>aces in general <lo not determine one magnitude 



only. Take, for instance, two lines in s|)ace. They have a 



di-ianre, and form an angle. If their distance or the sine i>f iheir 



i- = o. they •\ill Ik; coplanar. If Ijoth arc = o, they will 



• . We have two magnitudes, because the system of two 



■ h.xs two degrees of degeneration (coplanarily and 



' This is also generally the ca.se, lieraust- geo- 



" ;Miiudcs are nothing but ihe mosl suitable invariants, 



whoM: eiancscence is the necessary and sultirienl condition for 



(h.-d.-gon"r;(>i..n ..f ilv system lo which ihey belong. 



' A, B determine >m\y our magnitude, we 



M'-). Ix't .\ Ik- a straight line, for in- 



•"" • ■ ■" " '■ •' 1 I I", ^ince which has tine |Hiiiil in common with 



.\. Iroiii any [K^iinl of A, say I', draw the |M,-r|)cndicular t<i B. 



Join l!«iih point i,i, common to Ban<l A. Then the sine of . (J 



NO. 1345. VOL. 52] 



N. 



, , , , - , are obviously collinear. 



a - b h - ( (a 



is the magnitude denote<l by (.\B). Let .\ be a plane, having in 

 common with B a line. Krom any point P of the plane draw 

 the perpeiulicular on B, say PB, and from this jioint B tile per- 

 pendicular on the common line B(^). Then again sin ( . i)) 

 = (.\B), and thus generally. We determine the sign of the 

 magnitude according to the rule 



(.\B) + (BA) = o. 



Let us now add another plane s|->ace C to the systetn .\, B, such 

 that both C.V and CB determine only one magnitude. Then 

 the whole system may determine an additional one, whose evan- 

 escence would signify that C belongs to the )ilane space fixed by 

 A'.and B in conjunction, and is unired with the space that .\, B 

 have in common. It is in fact the product of (.\B) and the 

 m.agnitude formed by C and the space .\B, and will be written 



(ABC) 



In this w.iy we proceed, obtaining the definition of a m.tgnilude, 

 which has the property that its evanescence is the necessary and 

 sutticient condition for the ilegeneration of the system to wliich 

 it belongs. 



The magnitude in questiun may be formed in various ways, 

 hut the system being such that it can possess only one such mag- 

 nitude, the diflerent formations must always lead to one and the 

 same result, with the exception of a constant factor. This factor 

 must either be -^ I, or else - i, on account of the symmetrical 

 way in which the magnitude is formed. If the system is one of 

 straight lines through a ]X)int P, the magnitude in question has a 

 special significance. Two triangles which ha\e an angle in 

 common, are in proportion as the proiluct of the sides including 

 this angle. Three lines in sjxice which hiive a point in connnon 

 and are not coplanar, form a corner. Cut a corner by two ilif- 

 ferent planes. The two dift'erent pyramids are in proportion as 

 the product of the three sides forming the corner. .\nd so in 

 general, as can be easily i>roved by induction. Therefore, if we 

 have such a corner of 11 lines in a space S„ and cut it by a space 

 .S„., the pyramid formed is = the product of the n si<les exk-nd- 

 ing from the vertex of the corner multiplied with a factor which 

 is specific for the corner ; and this latter factor is exactly the 

 magnitude formed accoriling lo the rule given. 



(It may happen that the formation of the magnitude, as given, 

 leads to zero without giving a significant result. This is 

 an indication that somewhere during the process one of the 

 conditions of degeneration is fulfilled — for instance, when C 

 belongs to ihe space .\B. T'len the process is the reciprocal 

 one. We determine the magnitude formed liy C and the space 

 connnon to A and B. If that also is zero, then i\, B, C belong 

 to what is called a pencil. The simjilest case of this Uiml is 

 the system of three Imes in a plane. ) 



Let .\ B t" be three plane spaces belonging to a pencil : ihai 

 is, let (.\BC) = o. Let U be any other plane sjMCe, which has 

 an efemenl with the jjencil in common. Then we have again 



(AB)C + (BC)A -f (CA)B = o, 



where the single letters .\, B, C in this identity denote the 

 magnitude formed between each of these three S|)aces and the 

 auxiliary one. 



It will suffice lo imive this for the case of three lines through 

 a point P. Let 2 cut the pencil in a line S. Let .V, B, I" form 

 with .S the angles a, /3, 7 respectively, (hen the proposition 

 amounts to 



sin (a-/8) sin y -t- sin (fl-7) sin a + sin (7- a) sin fl = o, 



which is nothing but the Plolemiius theorem about four points 

 in a circle. 



Now again we may proceed to .show, that between // -I- 2 

 elements .\,. for which, to be short (.\^.\^ . . . .\„ + ») 

 = o, a linear relation must exist 1 a, Ai = o, where the 

 a/ arc certain constants. Of course, if not also some of ihe 

 minors are zero, such as (A, A, . . . A/i-t-,\ this will be the 

 oii/y relation that can thus exisi. We can therefore iletermine 

 the (I,, by giving 2 exceptional positions. The result is again 



(AjAj . . . A,, .(- ,) A,i + 5 -f (AjAj . . . .\„ + , A„ -)- ,) A, 



( (.Aj. . . A« I jAiJ.Vj + . . . = o. 



Lei Aj . . . A« f , form the space S, and the magnitude (.S) 

 then, making 2 identical wilh S, we obtain 



(.\,,S). (S)A, + (.SA,).(S).\, = o. 



But (AjS) = - Ajfor this special position of 2, and (S .\,) = .\|, 

 therefore Ihe test applies, and the theorem must be correct. 



