64 ON TRILiINEAR CO-ORDINATES., 



since {vide Can. Jour. 1865) sin '^d^ sin 2.4 + ... = 2 sin A sin 5 sin C. 

 It may be noted that 



{bn — cmy&va2A-{- ... = 4 A |^^ + 2raracos^ ~ ... | 



And 



Z cos 9^ + 7?i cos 0^ 4- n cos Qg = JZ^^ ... — 2mncos J.— .... j^ (6) 



5. If two lines (J, m, n), (I', m', n') are parallel, it follows from (5) 



that 



hn — cm cl — an am — bl 



bn^ — cm^^ eV — an^^ am' — hV 



6. To find the condition that the two lines (l, m, «), Q' , m', n') shall 

 be perpendicular to each other. 



From the relations (4) and (6) we get 



I sin 9 -]- m sin 9^ -\- n sin 63 = 



I cos 9 -{-m cos 9^ + n cos 9„ = R, suppose 



Z''cos 9 + 7n''cos 9^-\- n'cos 9^=0 



Z^sin 0^ -}- 7re'sin 0^ + ?2^siQ e^ = J?^ 



Multiplying the 1st and 4th of these equations together, and also the 

 2nd and 3rd, and adding the resulting equations, we get 



IV + ... + {m'n -\- mil') cos 0^^^+ ... = 0, 



or 



W -\- ... — [m'n 4- vin')cos .4 — .,. = 0, 

 the required condition. 



7. To find the length of the perpendicular p from (/, g, h) on 



0, m, «■)• 



The equations to the line through (/, g, k) perpendicular to 

 (J, m, n) are 



cos 0^ cos 0^ cos 0^ 

 ^Z(a-/)+... 



^COS0'' 4- ... ' 



1 ' 



I cos 9^-{- m cos 00+^ cos 0^ 



y+ inff 4- nh 



{ p-^ ... — 2mwcos^ — ...}^ 

 The relation (3) follows at once from this. 

 8. To find the distance d between the points (a, /?, y), (a', /9', /). 



