VOL. L.] PHILOSOPHICAL TRANSACTIONS. 21 J 



makes the SQth of Keill's Trigonometry, it appears, that if ac, am, be two arcs, 

 then sin. 4- (ac + am) X sin. 4- (ac — am) = {b -]- d) X (b ^ d) =: (sin. ^ ac 

 + sin. 4- am) X (sin. -i- ac — sin. -i- am. And in the solution of Case 2, the first 

 of these products will be the most readily computed. 



Case A. When the part required stands opposite to a part which is also un- 

 known : having from the data of Case 1 found a 4th part, let the sines of the 

 given sides be s, *; those of the given angles 2, o-; and the sines of half the un- 

 known parts a and b\ and we shall have, as before, ssa^ + cP — Z;^ =: 0; and if 

 the equation of the supplements be So-a'^ -(- ^^ — (3* = 0; then, because a^ = 1 

 — ^^ = 1 — {ssci^ -|- d^), and (3'^ = 1 -- a^, substituting these values in the 2d 

 equation, we get 



Iheorem 3. , — ^ — — = a^ ; m words thus : 



Multiply the product of the sines of the two known angles by the square of the 

 cosine of half the difference of the sides : add the square of the sine of half the differ- 

 ence of the angles ; and divide the complement of this sum to unity, by the like 

 complement of the product of the 4 sines of the sides and angles ; and the square 

 root of the quotient shall be the sine of half the unknown angle. 



If we work by logarithms the operation will not be very troublesome; but the 

 rule needs not be used, unless when a table of the trigonometrical analogies is 

 wanting. To supply which, the foregoing theorems will be found sufficient, and 

 of ready use ; being either committed to memory, or noted down on the blank 

 leaf of the trigonometrical tables. 



These theorems Mr. Murdoch demonstrates. 



LXXL Two extraordinary Cases of Gall-stones . By James Johnstone, M.D.* 



of Kidderminster, p. 543. 



The calculus voided in the 1st of these cases was of a pyriform shape, re- 

 sembling the form of the gall-bladder. It was smooth and polislied, except at 



* For the following particulars respecting the author of the above paper, we are indebted to Dr. 

 John Johnstone, an eminent physician at Birmingham, one of Dr. James Johnstone's sons. We 

 regret that the limited space allotted to biographical notices would not admit of our inserting more 

 than an abstract from Dr. John Johnstone's more extended and highly interesting account of the 

 life and writings of his late father. 



James Johnstone, m.d. who practised physic for half a century with great reputation at Worcester, 

 and distinguished himself by several ingenious works, was the 4th son of J. J. Esq. of Galabank, an 

 ancient branch of the Johnstones of Johnstone. He was born at Annan, April 14, 1730, and in that 

 town imbibed the rudiments of his scholastic education, under the Rev. Dr. Robert Henry, afterwards 

 celebrated for his History of Great Britain. He studied physic at Edinburgh under the elder Monro, 

 and the Professors Whytt, Rutherford, St. Clair, and Plummer, and there was admitted doctor in 



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