DR MATTHEW STEWART'S GENERAL THEOREMS. 591 



p . S (am p m cos 3 6 m ) = S a m . (q p cos 3 u p ) ........ (11) 



p . S (a m p m sin 3 OT ) = S a m . S (q p sin 3 W P ) ........ (12) 



p . S (a m cos 4:6 m ) = S a m . S (cos 4: u p ) ......... (13) 



p . S (a m sin 4 m ) = S a m . S (sin 4 ,) ......... (14) 



which giving fourteen independent equations furnishes data for finding seven lines, 

 instead of Jive, as stated by Dr STEWART : that isp=7, instead ofp5. 



The preceding statement is that of Prop. 38, of which the others are parti- 

 cular cases. It becomes, Prop. 37, when a l a^= .... = a m . When all the lines 

 are parallel it becomes the first case of Prop. 36. In this instance we have 



6 1 = 6 2 = ....== Q m = TT. 







Substituting these values in the preceding equations, we have, by transposi- 

 tion, 



) ......... (15) 



*) .......... (16) 



Sa m . Sfap 3 cos w p ) =0 .............. (17) 



S a m . S (g p 3 sin w p } =p . S (a m p m z } ......... (18) 



Sa m . S(y p cos w p ) =0 .............. (19) 



S a m . S (y p sin w p ) p.S(a m p m } .......... (20) 



Sa m .S(q p *cos2 Wp) = -j. S(a m p m *} ......... (21) 



Sa m . S(y P 2 sm2 Up)=Q .............. (22) 



S a m . S (cos 2 u p ) =p.Sa m ........... (23) 



a OT ./S(sin2 Wp) =0 .............. (24) 



Sa m .S(y p cos 3 w p ) =0 .............. (25) 



Scm . (? P sin 3 w^) =-p.S(a m p m ') . ....... (26) 



Sa m . S(cos4 w p ) =p . S a m ............ (27) 



S a m . <S(sin4 w p ) =0 .............. (28) 



Now of these fourteen equations, eight (viz. 17, 19, 22, 23, 24, 25, 27, 28) are 

 fulfilled by w 1= w 2 = ---- = ca p = - -TT, giving the lines to be found, parallel to the given 



lines. The other six equations are, with this condition, reduced to the four fol- 

 lowing ones : 



from (20, 26).... ,.(&) =p.S(a m p m ) ........ (29). 



from (16) ---- Sa m .S(g p *)=p.S(a m p m *) ....... (30) 



from (18, 21) ---- S a m . S (y p 3 ) =p . S (a m p m 3 ) ........ (31) 



from (15) ---- Sa m . S($- p 4 )=/>. S (a m p m *~) ........ (32) 



In this case we have j?=4, instead of p=2, as stated by Dr STEWAHT. 



[See also, NOTE E.] 



