2 REPORT— 1861. 



plicated technical papers whicli could not be understood without a month's study 

 of a printed book. In the next place, recapitulations of what had been done before 

 ouo-ht to be as brief as possible. And whilst he hoped that the papers would be 

 well discussed, he trusted that personality of every kind woidd be strictly eschewed. 

 It was known to those present that great ingenuity had been employed upon cer- 

 tain abstract propositions of mathematics which had been rejected by the learned 

 in all ao-es, such as finding the length of the circle, and pei-petual motion. In the 

 best academies of Em-ope, it was established as a rule that subjects of that kind 

 should not be admitted, and it was desirable that such communications should not 

 be made to that Section, as they were a mere loss of time. 



The President then stated that at the last meetiug Professor Stokes was requested 

 to make a repoii, at the instance of the Committee of that Section, on " The present 

 state of Physical Optics." He had written to say that he had been prevented by a 

 gi-eat quantity of public business from preparing a report in time for that meeting, 

 but he engaged to prepare it in time for the next meetiug of the Association, and 

 the Conuuittee had requested him to do so. He had to make a similar explanation 

 with reference to a report that had been promised by JNIi". Cayley on the solution of 

 specific problems of dynamics. The Committee had requested him to prepare his 

 report in time for the next meeting of the Society. « 



On Curves of the TJiird Order. By A. Catlet, F.B.S. 

 A cirrve of the third order, or cubic curve, is the section of a cubic cone, and 

 such cone is intersected by a concentric sphere in a spherical cubic. It is an obvious 

 consequence of a theorem of Sir Isaac Newton's, that there are five principal kinds 

 of cubic cones, or, what is the same thing, five principal kinds of spnerical cubics ; 

 but the natm'e of these five kinds of spherical cubics was first distinctly explained 

 by Mobius. They may be designated the simplex, the comphx, the crunodal, the 

 acnodal, and the cuspidal; where crimode, acnode denote respectively the two 

 species of double points (nodes), viz. the double point with two real branches, and 

 the conjugate or isolated point. The foregoing results are known; the special 

 object of the paper was to establish a subdi\-ision of the simplex kind of spherical 

 cm-ves. The simplex kind is a continuous re-entering cmne cutting a great circle 

 (to fix the ideas say the equator) in three pairs of opposite points, which are the 

 three real inflexions of the cm-ve. The three gi-eat circles, which are the tangents at 

 the inflexions, and the equator, divide the entire surface of the sphere into fourteen 

 regions, whereof eight are ti-ilateral, and the remaining six quadi-Uateral. The cur^'e 

 may lie entirely in six out of the eight ti-ilateral regions, and it is in this case said 

 to be simplex trilateral ; or it may lie entirely in the six quadrilateral regions, and it 

 is in this case said to be simplex quadrilateral : and there is an intei-mediate foiTU, 

 the simplex neutral; \az. in this case the three great circles, tangents to the in- 

 flexions, meet in a pair of opposite points, and there ai-e in aU only twelve regions, 

 all trilateral ; the curve lies entu-ely in six of these regions. 



On t7ie General Forms of the Symmetrical Properties of Plane Triangles. 

 By Thomas Dobson, B.A. 



This paper establishes among the distances from an indefinite plane of the sym- 

 metrical points connected -with a plane ti-iangle certain genei-al relations, which yield 

 several con-esponding cognate plane properties when difterent definite positions are 

 assigned to the plane of reference. In the plane triangle A B C, let O be the centre 

 of the inscribed circle, O^ Oo O3 those of the escribed cii-cles touching BC, CA, AB ; 

 and let 00^ cut BC in D. Denote, as usual, the radii of these circles by r r.r^ r^, the 

 sides opposite to A, B, C by a, J, c, and a + 6 + c by 2s. Fi-om A, B, C, 0, Oi, 0^, O3, 

 and D let perpendiculars u, /3, y, 8, h^, 8^, 83, and m be dra-wn to any plane. 



Then, hj similar triangles, 



m-cc AD_ 2s^' y-m _CD_6 Si^i^_^i__!_ 



8-«~AO~6 + c' jn-i3~BD~c' ^^"^ b -cc AO s-«' 

 eliminating m, Sec, 



2(8=a«+?i3+c-y ; 2 (s-a) 8, = -a»+ip+cy; 



2 (s-b) d,,=ac(-b^+cy; 2(s-c)83 = ax+b^-cy. 



