REGULAE FOUR- DIMENSIONAL HYPEESOLIDS. 13 



15. Let the GOO-cell be cut by a space ^ throu2;h A B C D 



3 ■( 



parallel to S ; I{A ) intersects S in a plane through A parallel 



.3 



to its intersection with S . 



5 



In fig. li) this plane is projected in the line A V and comparing 

 this with D' r we have the points (Aê)^^ {D\), (Aê)^^ {D' ^), U" B) ^^ {A Ö), 

 {A' C) {A Ô) (diagram VI points 1, 3, 4, 2, 5). Here J S and S\ 

 intersect in a plane parallel to {B' ) {D' ) {B' ) through (Aê) {!)' ). 

 This plane is projected in the line r v , fig. 20, where r (Aê) ^pa. 

 Then {B' ) V ^= a times the edge, whence we have the points 



(X>' ) {t ci),{B' ^){h''^),{B\)^^ [ty) (diagram YI points 12,11,13). 



Now >S' and / -/ intersect in a plane parallel to iA ) {A ) {A ) and 



passing tii rough the points (A'B) {.le). 



In fig. 22 the line {A'B){Aè) is divided in the ratio a-. 1 — a 

 and through the point of division a line r V is drawn parallel to 



{A ){A ). 



Let {A'B)r = m times the edge, and it wUl be found that {Aê) {A ) 

 is divided in the ratio // : 1 — //, {AAJ) {Ad) is divided in the ratio 

 a : 1 — a, {A'C) {a ) is divided in the ratio m : 1 - - v/, {A'C) {A /3) 

 is divided in the ratio a : \ — a and finally V {A j2) = // times the 

 edge ; whence we have the points 



{A'B) {a), {£-B) {Ah {IB) {Ay)^ {Aè){A), 



III p a (I ' il IJ 



{Ay) {A), {AX') {Aè)^ {/B) {Ay), {/ B) {u^). 



Il C II II ' III à 



{A'C)^Ja^), (jV')^^ (.7/3), {A'B)^^{A(2), U /3\, (-/> 

 Here S' and J {A S) intersect in a ])lane through A })arallel to 

 the intersection of &\ with 1 {A è); A r, fig. 21, is a projection 

 of this plane. Let {B ) r = jj n. It will he found that {A' C) {a! ) 

 is cut in tiie ratio m: 1 — in, {A' C) (tJ' |3) is cut in the ratio c: 

 1 — c, {B' ) {è /3) is cut in the ratio a-. ] — a, whence we have 



the pomts {A" C)^^^ {u^), {/B)^^^ [a'^), {/C)^ {f (i)^ {A" B)^ {f y), 

 {B'J^^ {f /3), {B\)^^ {fy), {B'^)^^ cJ^, (D;.)^^ ê^. These are, diagram 

 VI, the points 15, 22, 24, 23, 11, 13, 20, 25, see list. 



