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... INATION QUESTIONS AND PROBLEMS. 257. The points P, Q, R, S are taken on two orthogonal diameters of a circle, centre 0, so that OP= OQ= OR=OS; AB, BC, CD, DA are the polars of the points. AP produced meets BR in p; prove that the locus of p and the analogous points is a cubical parabola, and that AP, BR and the analogous lines envelope hyperbolas. R. TUCKER. 258. Every number can be expressed as the sum of 1, 2, 3, or 4 squares. How many numbers are there less than 1000 which cannot be expressed as the sum of 1, 2, or 3 squares. W. ALLEN WHITWORTH. 259. There is no number of six figures which does not contain factors prime to 11 111 1. W. P. WORKMAN. SOLUTIONS. A great number of solutions are in hand, and will be published as sufficient space is available. Solutions are wanted for Nos. 90, 140, 150, 10" 171,172, 194, 219 (for write c 236, 238. ERRATA. p. 166, line 26: for 'mq' read ' nq.' p. 171, line 12: for 'p/m3' read ' pB/m3.' " " 13: for 'p'insert'pc.' 257. The points P, Q, R, S are taken on two orthogonal diameters of a circle, centre 0, so that OP= OQ= OR=OS; AB, BC, CD, DA are the polars of the points. AP produced meets BR in p; prove that the locus of p and the analogous points is a cubical parabola, and that AP, BR and the analogous lines envelope hyperbolas. R. TUCKER. 258. Every number can be expressed as the sum of 1, 2, 3, or 4 squares. How many numbers are there less than 1000 which cannot be expressed as the sum of 1, 2, or 3 squares. W. ALLEN WHITWORTH. 259. There is no number of six figures which does not contain factors prime to 11 111 1. W. P. WORKMAN. SOLUTIONS. A great number of solutions are in hand, and will be published as sufficient space is available. Solutions are wanted for Nos. 90, 140, 150, 10" 171,172, 194, 219 (for write c 236, 238. ERRATA. p. 166, line 26: for 'mq' read ' nq.' p. 171, line 12: for 'p/m3' read ' pB/m3.' " " 13: for 'p'insert'pc.' 131. Prove geometrically that sin 3A = 4 sin A sin (60? + A)sin (60 -A).