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which we denote by Let OPA (Fig. 1) represent aA and OAQ represent B; then OPQ, the third side of the spherical triangle, represents the product aAB. To prove that aAB = cos A cos B - sin A sin B cos a + {cos B sin A a+cos A sin B - sin A sin B sin a}. The first part of this proposition, namely, that cos aAB = cos A cos B - sin A sin B cos a, is...
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which we denote by Let OPA (Fig. 1) represent aA and OAQ represent B; then OPQ, the third side of the spherical triangle, represents the product aAB. To prove that aAB = cos A cos B - sin A sin B cos a + {cos B sin A a+cos A sin B - sin A sin B sin a}. The first part of this proposition, namely, that cos aAB = cos A cos B - sin A sin B cos a, is equivalent to the well-known fundamental theorem of Spherical Trigonometry; the only difference is, that a denotes, not the angle included by the sides, but the angle between the planes; or, to speak more accurately, the angle between the axes a and. It is more difficult to prove the complementary proposition, namely, that Sin aAB = cos B sin A a + cos A sin B - sin A sin B sina a, for it is necessary to prove, not only that the magnitude of the right-hand member is equal to - cos2aAB, but also that its direction coincides with the axis normal to the plane of OPQ At page 7 of Fundamental Theorems, I have proved the above statement as regards the magnitude, but I was then unable to give a general proof as regards the axis. Now, however, I am able to supply a general proof, and it will be found of the highest importance in the further development of the analysis. In Fig. 1, OP is the initial line of aA, and OQ the terminal line of B; let OR be drawn equal to cos B sin A a + cos A sin B - sin A sin B sina a; it is required to prove that OR is perpendicular to OP and to OQ. Now, OP = a-A = (cos A - sin A). About the Publisher Forgotten Books is a publisher of historical writings, such as: Philosophy, Classics, Science, Religion, History, Folklore and Mythology. Forgotten Books' Classic Reprint Series utilizes the latest technology to regenerate facsimiles of historically important writings. Careful attention has been made to accurately preserve the original format of each page whilst digitally enhancing the aged text.
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