148 A NEW METHOD OF DETERMINING 
The transformation should be carefully checked by being done in duplicate, or 
better by putting the angle 7g = 0, in all the divisions of the two functions, having 
thus only the angles (0 — #’), (0— 2H’), (0 —3#’), ete., ete.; also (0 — g’), (0 — 
2g’), ete. Adding the coefficients in each division of the functions before and after 
transformation, and operating on the sums before transformation as on single members 
of the sums, the results should agree with the sums of the divisions of the transfor- 
mations given above. 
The transformations of these functions were checked by being done in duplicate, 
but we will give the check in case of another planet. We have for the logarithms of 
the sums before transformation, and for the sums after transformation the following : 
g Ja! cos sin Ge af COS sin 
0—1 1.85407 1.62090n 0—1 + 10.548 — 40.188 
0— 2 1.25778 1.51473n 0— 2 + 19.809 — 32.318 
0— 3 9.7024n 1.26993n 0 = 8 + 0.906 — 19.852 
0—4 0.71012 0.9147n Q0—4 — 4.540 — 9.268 
C= 5) 066327) 0:3899n 0=h = Ary = B33 
0 — 6 0.4387n 9.0934 0 — 6 — 3.059 — 0.330 
0—1 0.1222n 9.8069 0—7 — 0.623 + 0.739 
0—8 9.5965 9.8865 0—8 -—— 0.071 + 0.615 
For the angle (0 —1), (0 — 2), OQ = 8. 
1) ey Ww 1) 
— 0.041 + 0.024 + 1.722 — 1.007 + .062 —  .087 
— 0.873 + 1.578 — 042 + .076 + 871 — 1.574 
.000 — 0016 + .037 + 1.346 + .003 + .097 
+ 71.462 — 41.774 — 012 — .O19 + 494 + (791 
+ 70.548 — 40.188 + 18.104  — 32.714 — .020 — .O11 
+ 70.573 — 40.196 + 19.809 — 32.318 — 504 — 18.618 
fae + 19.811 — 82.319 + 0.906 -+ 19.852 
Bene hs, + 0.902 - — 19.355 
The numbers in the last line of each case are the sums of the divisions after con- 
version when 7g is put = 0. 
