Several constitutions will not receive infection, let them be inoculated ever so judiciously. A Ranby, a Hawkins, a Middleton, and other inoculators, will tell us, that the incisions have sometimes suppurated so much, and pustules have appeared round the edges of the wound, without any other particular marks of the disease; and yet the patient has never had the small-pox afterwards. The Marquis mentions an instance somewhat of the same kind in his first Memoir, p. 147. The examination of these very important and interesting particulars has, I observe, drawn me into a prolixity, which I fear may prove tedious to your Lordship: but should I have removed all doubts, and brought convincing proofs of the absurdity of fearing a second infection; should I have shewn inoculation to be a necessary practice, and that the contagious distemper may be communicated more ways than one; I hope your Lordship will excuse the length of this letter. I shall only add my earnest wishes, that the legislature may, by effectual means, prevent the importation of distempered cattle and hides into these kingdoms; the only means of naturalizing and perpetuating a dreadful distemper, now, thank God! much decreased among us. I am, with the greatest respect, My Lord, Your Lordship’s Most humble and most obedient Servant, Daniel Peter Layard. Huntingdon, 26 Nov. 1757. LXX. _Trigonometry abridged. By the Rev._ Patrick Murdoch, _A. M. F.R.S._ [Read Feb. 2, 1758.] THE cases in trigonometry, that can properly be called different from one another are no more than _four_; which may be resolved by _three_ general rules or theorems, expressed in the sines of arcs only; using the supplemental triangle as there is occasion. [Illustration: _Philos. Trans. Vol. L._ TAB. XX. _p. 539_. _J. Mynde sc._] CASE I. _When of three given parts two stand opposite to each other, and the third stands opposite to the part required._ THEOREM I. _The sines of the sides are proportional to the sines of angles opposite to them._ DEMONSTRATION. Let QR (TAB. XX. _Fig._ 1.) be the base of a spherical triangle; its sides PQ, PR, whose planes cut that of the base in the diameters QC_q_, RC_r_. And if, from the angle P, the line PL is perpendicular to the plane of the base, meeting it in L, all planes drawn through PL will be perpendicular to the same, by 18. _el._ 11. Let two such planes be perpendicular likewise to the semicircles of the sides, cutting them in the straight lines PG, PH; and the plane of the base in the lines LG, LH.