370 Defence of English Periodical Mathematical Works. 



of the foci. On one side of the house he can walk at the rate 

 of five miles an hour ; but, on the other, only four miles an hour. 

 Required the least time in which he can perform his rectilinear 

 journey across the common. 



9. Demonstrate, synthetically? that an arc of any curve cannot 

 be of finite curvature, unless the subtense ultimately vary as' the 

 square of the arc. 



10. In order that the eclipses of Jupiter's satellites maybe vi- 

 sible at any place, the planet must be at least 8" above, and the 

 sun 8° below, the horizon. Now, on the 1st of February 1821, 

 there will be an eclipse of .Jupiter's third satellite, the emersion 

 taking place at 6^ 19™ 3^ P.M. Greenwich time: can that emer- 

 sion be observed at Berlin, N. lat.52-' 32' 30'', E. Ion. 13° 26' 15"? 

 Sun's declin. S. 17° 6'. Jupiter's declin. S. 3° 31'. Passage 

 merid. 2^ 39" P.M. 



11. Suppose BC to be perpendicular to the horizontal line AC, 

 and both of them to be given in length, and that a string of a 

 given length (greater than '^/(AC^ + BC') is fastened at its two 

 extremities to the points or tacks A, B ; and that the said string, 

 as it is moved round A, is stretched tight by the continual mo- 

 tion of a point along the horizontal plane. Required, the curve 

 that will thus be described on the horizontal plane, by the said 

 stretching or tending point, 



12. What will be the ratio of the forces of gravity, at the sur- 

 face of the earth, at the top of a slender cylinder a mile high, at 

 the upper surface of a sphere a mile in diameter (in contact with 

 the earth and of the same medium density), and on the top of an 

 extensive piece of table-land of the same density and a mile high, 

 taking the earth's radius at 3960 miles ? 



13. In the great solar eclipse which happens Sept, 7th, 1820, 

 the apparent time of the greatest obscuration at Greenwich is 

 Ih 52'" 48''2 P.M. when the angular distance about the centre 

 of the sun from its vertex to the centre of the moon is 17° 18' 22" 

 to the left hand ; and the visible distance of their centres to the 

 difference of their horizontal parallaxes as 0*5665946 to 1, Hence 

 it is required to find the direction and distance from the above 

 place, which a person must travel for the shortest joz/rwey to ob- 

 serve the sun centrally eclipsed ; and at the same time the angle 

 of elevation, and the direction he must ascend in a balloon for 

 the shortest voyage to be gratified with a sight of the same phae- 

 nomenon ; supposing the earth a perfect sphere, and its diameter 

 7914 miles ? 



14. It is required to investigate a theorem, comprehending the 

 pressure on the base of a steam-engine piston, the course or stroke 

 of that piston and its velocity, on one part; and the velocity, 



mass. 



