Septembbe 27, 1895.] 



SCIENCE. 



395 



4. On a generalization of Weierstrass' s equation with 

 three terms : Peof. F. Moeley. 



5. Formulas for the sides of rational plane triangles : 

 De. Aetbmas Maetin. 



6. Partial linear transformations of ternary quantics and 

 their concomitants : PROF. J. McMahon. 



7. An introduction to the integrell calculus : PeOF. \V. 

 H. Echols. 



8. On the expansion of a uniform function of a real vari- 

 able without use of derivatives : Peof. W. H. 

 Echols. 



9. On continuous functions without differential coeffi- 

 cients : Me. p. a. Lambert. 



10. Concerning Jordan's linear substitution groups : 

 Prof. E. H. Mooee. 



11. Algebraic symbols and j/ — 1 : Peof. A. L. Baker. 



12. An application of the method of conformal repre- 

 sentation to the study of related differential equations : 

 Peof. E. B. Van Vleck. 



13. On the differential equations of certain systems of 

 conies : Mr. E. A. ROBERTS. 



14. On bilinear forms : Peof. H. Tabee. 



15. Elementary proof of the quarternion associative 

 principle: PROF. A. S. HATHAWAY. 



16. Asymptotic lines on. a circular ring : Peof. H. 

 Maschke. 



Dr. Hill's paper at the time of its presen- 

 tation was already in type for publication 

 in the Astronomical Journal. Its object is to 

 obtain the values of the coefficients of the 

 periodic inequalities having the multiples 

 of the mean angular distance of the Moon 

 from the Sun as arguments when the in- 

 clination of the lunar orbit and the two ex- 

 centricities are neglected. It is very de- 

 sirable to have these coefficients with a high 

 degree of accui-acj^ in order to effect their 

 useful employment in the further determina- 

 tion of the motion of the perigee and node, 

 and in fact of all the other coefficients of 

 the periodic inequalities. This work has 

 been done previously by the author in the 

 Ameriaan Journal of Mathematics, Vol. I., but 

 in neglecting the lunar mass and the solar 

 parallax. Mr. Ernest W. Brown has in the 

 same journal supplemented these researches, 

 but still leaving out of consideration the 

 mass of the Moon. 



Professor Shaw's first paper is a develop- 

 ment of the linear vector operator of qua- 



ternions in the form of a scalar part and 

 two vector parts, also as a tensor and a 

 versor, and finally as a sum of nine operators. 

 The forms are similar to those obtained in 

 the articles of Professor Taber (Amer. Jour. 

 Math., Vols. 12, 13), but are here developed 

 entirely from quaternion expressions and 

 not from matrices. It was stated that this 

 paper would be offered to the American 

 Journal of Mathematics for publication. 



Professor Shaw's second paper applies the 

 quaternion calculus to homogeneous geome- 

 try of two dimensions. In the expression 

 ^ = xi -\- yj -^ zk, X, y, z, are proportional to 

 the areas PBC, PC A, PAB respectively, 

 where the point P is referred to the funda- 

 mental triangle ABC. By this convention 

 propositions of projective geometiy are 

 easily proved, and especially such proposi- 

 tions of modern geometry as those of the 

 Lemoine-Brocard type. The author ex- 

 pected to contribute this paper to the Annals 

 of Mathematics. 



Professor Morley's paper will be published 

 in the Bulletin of the American Mathematical 

 Society. It contains a simple generalization 

 of the formula 



(T (tt+Mi) O itl—Ui) O (M2 + «3) a («2— "s) 

 + C (m-I-Mj) " (« ^) " ("3+'*l) " (% "l) 



+ (T ('M-I-M3) cr {u — M3) a (Mj-(-1(.2) n (mj — u.^^^0 



In Dr. Martin's paper a large number of 

 formulae are deduced for calculating rational 

 numbers which represent the sides of tri- 

 angles having a rational area. Among 

 them are, for the case of a right angled 

 triangle, 



x=2pq, y^^p'' — 2^, 2=p^-)-2^, 



and, for the case of an oblique triangle, 



x={p^+q^) [r^—s^), 



y=Srs ip'+q'') ±Ss^ {p''—f), 



Z=(p2+g2) [r^-i-s''), ±3rs (p^^q^), 



in which x, y, z denote the sides, and p, q, 

 r, s, any entire numbers whatever. The 

 paper, which contains many numerical ap- 

 plications, will be printed in the Mathematic- 



