SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1047 



In the next example I shall take the assumed quantities £1,2... 171,2... from the trigono- 

 metrical tables; so that log^^logcos X ; log ^ = log sin X ; and so on, and thence 

 determine x 1>2 . . y Xt2 ... for parameter = 1. 



Example 2. 



Values of x and y will be found to the arguments, = 20°; = 30°. 

 1. Let0 1 = 2O°; £i = cos 20°; 17! = sin 20°; x 1 = ^ 1 /w 1 ; yi = r}ijwi. 



Log cos 20° 

 log cos 6 



= 1-9730 

 1-8380 



1-7840 

 1-7300 



= 1-5341 

 10682 



21364 

 53410 



= 1-7907 

 1-9791 



lo gfi = 

 -\ogw 1 



£2 = cos 30 



= 1-9375 

 T-6250 



T-5000 

 T-3750 



T-6990 

 T-3980 



2-7960 

 4-9900 



T-5348 

 1-9534 



log | 2 = 



— log w 2 



\ogx 2 



= 1-9730 

 1-9791 



Nat. numbers. 



|V = 00712 

 | 10 05370 



IV 00094 

 >; 10 (insensible) 



w 10 = 0-6176 



log cos 8 =1-7840 

 log sin 2 +1-0682 



log cos 8 

 log cos 10 



Log sin 20° 

 log sin 2 



2-8522 

 = log |V 



log cos 6 =1-8380 

 log sin* + Tl364 



log sin* 

 log sin 10 



log w 10 



3-9744 

 = log |V 



log w 



log Vl -- 

 -logw 1 



30°; x 2 = % 2 /w 2 ; y 2 =rj 

 Nat. numbers. 



|V = 00791 

 I 1 0-2371 



|V 0-0264 

 »; 10 (insensible) 



w w = 03426 



= 1-5341 

 1-9791 



2. Let = 30°; 



Log cos 30° 



log cos 6 



1-9939 

 '°; 172= sin 



= 1-9375 

 1-9534 



1-5550 



2 /w 2 . 



log cos 8 =1-5000 

 log sin 2 +1-3980 



log COS 8 

 log COS 10 



Log sin 30° 

 log sin 2 



2-8980 

 = log |V 



log cos 6 T-6250 

 log sin* +2-7960 



log sin* 

 log sin 10 



log w 10 



24210 

 = log |V 



log V) 



log*7 2 = 

 log w 2 



logy 2 



= 1-6990 

 1-9534 





1-9841 



1-7456 



Resuits/^' 9860 ;^- 3589 ; 



U 2 = -9640; 2/ 2 = -5567 - 

 The following independent analytical proof of the general theorem, including its 



