the foregoing expression for tan. takes this very remarkable form, viz.

tan. Stan. ( S − a) tan. (Sb) tan. ( S − e);

which is Lhuillier's expression.

It follows from this problem that two spherical triangles are always equal in surface when the sides of the one are severally equal to those of the other, whether the triangles admit of coincidence or not.

PROBLEM IV.

Given the area of a spherical triangle on the surface of the earth in square feet, to determine the spherical excess.

Let the area of the triangle in feet be Σ, then, by problem 1.,

Now the length of a degree, supposing the earth to be a perfect

sphere, is 365154.6 feet; hence the earth's radius is

180

X

3.14159

Hence, from the logarithm of the area of the triangle in feet, subtract the constant logarithm 9.3267737, and the remainder will be the logarithm of the excess in seconds.

This rule, which usually goes by the name of General Roy's rule, is in fact due to the late professor Dalby, by whom it was communicated to the General, when engaged with him in conducting the Trigonometrical Survey. (See the "Life of Mr. Dalby," in Leybourn's Repository, vol. v.)

of

By means of the rule just given we may very readily compute the spherical excess, provided that we previously know the area of the triangle in feet. In trigonometrical surveying, the triangle on the surface of the earth, comprised between any three stations, is necessarily so limited a portion of the whole sphere that its area, computed as a plane triangle from the measured data, cannot be affected with any error of consequence. On this hypothesis, therefore, the area of the triangle may be determined by one or other of the methods in prob. III., last chapter, and thence the excess ascertained by the above rule. Should the excess, thus deduced, exactly equal the excess of the three observed angles above two right-angles, we may be assured of the accuracy the observations; but if they differ, the difference must be regarded as the amount of the errors with which the three observed angles are affected. If all of them were observed with equal care, so that there appear no reason why one should be more erroneous than another, the correction thus found must be distributed equally among them; but if it be suspected that one of the angles is less to be depended on than the others, then to this angle must be applied the greater part of the whole correction. The data being thus corrected, the required side or sides of the spherical triangle may be computed by the rules of spherical trigonometry; or the same object may be effected by plane trigonometry, with all requisite accuracy, provided we employ in the computation, not the corrected spherical angles, but these angles diminished each by one third of the spherical excess found as above, a truth which has been

SURFACE OF A SPHERICAL TRIANGLE.

199

established by Legendre, (See the Appendix to Brewster's translation of Legendre's Geometry.) Trigonometrical surveying is a very important application of the theory of trigonometry, but is too ample a subject to admit of being discussed in the present volume. The student will find a condensed account of these geodetical operations in the tenth section of Dr. Lardner's Trigonometry, and every requisite information in the Géodésie of M. Puissant and Colonel Mudge's account of the Trigonometrical Survey of England and Wales.*

Miscellaneous Expressions involving the Sides and Angles of a
Spherical Triangle.

(85). We shall terminate the present chapter by the insertion of a few general expressions, involving the three sides and the three angles of a spherical triangle. Those formulas which have already been given in the second part of the work, are amply sufficient for the solution of every case in spherical trigonometry, but the sides and angles of a spherical triangle possess many other remarkable relations which are often called in aid in higher investigations concerning the sphere. A few of these, therefore, it may be proper to give. Let s represent half the sum of the sides a, b, c, and S, half the sum of the angles A, B, C, of a spherical triangle; then, by multiplying together the expressions for sin. ¿A, cos. A, in art. (47), and those for sin. a, cos. la, in art. (49), and squaring the results, we have these equations;

sin.26 sin.2c sin.2 A=4 sin. s sin. (s--a) sin. (s—b) sin. (s—c)=4n2..(1) sin.2B sin.2 Csin.2a-4cos.Scos. (S-A)cos. (S-B)cos. (S-C)=4N2.(2). By multiplication,

sin. a sin. b sin. e sin. A sin. B sin. C 4 Nn.

(3).

* Some additional particulars respecting the spherical excess will be

found in the supplement.

But the first two factors of this expression are each of them the re

Substituting in (6) the value of N deduced from (5), and in (5) the value of n deduced from (6), we have, from the resulting equations, these expressions for n and N, viz.

expressions which are remarkable for their symmetry.

Again, referring to the expressions for sin. A, and cos. A, at (47), we see that

And by referring to the expressions for sin. a, cos. ža, at (49), we

see the truth of the following analogous equations; viz.

* This expression, as well as those marked 19, is usually given with an improper sign, viz. + instead of -, a mistake which seems to have arisen from confounding cos. S cos. S with √ cos. S x cos. S, which are, in fact, distinct expressions; the one being + cos. S, and the other cos. S. See the chapter on Imaginary Quantities, in Young's Algebra.

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