and the Method of determining their Depth. 497 



which is the depression of the side of the cavity reckoning from 

 the surface of the sun, will be equal to A.Cn, the heliocentric 

 distance of the spot at the point nearest the limb. Now this 

 angle is easily obtained from the measurement of the spot at the 

 margin itself; and this distance is its versed sine : K A= versed 

 sine K C n. Hence knowing, either from the Tables or from 

 direct measurement, the semidiameter of the sun, R, and the di- 

 stance of the spot from the limb, D, we have 



R-D = CK=:IlcosAC«, 



whence the angle sought, 



cosAC«= — ^ — . 



The measurement of the penumbra of the spot in a cross 

 direction will give p n, from which op may be calculated I'cadily 

 by means of the formula 



op = np tan onp = np tan ACw. 



This theory supposes that the spot is circular and symmetrical 

 in its penumbra ; and from the rare occurrence of such circum- 

 stances, it happens that spots are somewhat scarce which are suit- 

 able for its being bi-ought into play. When, however, we meet 

 with isolated spots that present themselves with a regularly 

 formed penumbra in the central portions of the disc, and the regu- 

 larity of which is preserved equally both above and below when 

 they have approached the edge, it seems one may feel assured 

 the error arising from this cause will not prove very considerable. 

 It is consequently necessary to keep an attentive watch for those 

 circumstances that are most advantageous ; and by making suf- 

 ficiently numerous observations, any accidental irregularity may 

 be got rid of. These favourable circumstances seemed to me to 

 present themselves in one of the two spots that made their ap- 

 pearance at the beginning of March last, one of which, when it 

 had approached the limb on the 8th of that month, presented 

 itself without a penumbra on the inner side, while on the pre- 

 vious days it had been observed to grow continually less. The 

 following elements for the calculation were obtained from obser- 

 vations :— Rome, mean time, 1858, 8th of Marcli, l^^ SO"". Di- 

 stance of the border of the spot devoid of penumbra, from the 

 margin of the disc, =33"0; breadth of the penumbra in a 

 direction crossing that of the preceding distance =12"'035 ; 

 breadth of the penumbra on the outer side towards the edge 

 = l"-35. With these data, employing the solar parallax and the 

 diameter of the sun, as given in the ' Nautical Almanac,' for the 

 period of the observation, we find the inclination of the sides to 

 the surface to be =10^, and the depth to be 0'37 of the radius 

 Phil. May. S. 4. No. 110. 8uppl. Vol. 16. 2 K 



