October 1, 1886.] 



KNOW^LEDGE 



359 



the region of vertical clouds of condensation, whose upper 

 surface is the photosphere, while its lower surface marks the 

 inside limit of the penumbra, is intrinsicall)- brighter and 

 hotter (because of the heat liberated in condensation) than 

 the i-egion of vaporous matter in which the clouds form, 

 and above which they are suspended. Now the vaporous 

 region nuppun' must possess considerable refractive power, 

 since it is dense enough to produce great absorption. Hence 

 we are not only justified in expecting, but compelled to take 

 into account a measurable deflection of the lines of sight by 

 which the various parts of the interior of a spot are seen, 

 when the spot is near the edge of the solar disc, and so 

 viewed aslant. 



Suppose now ^p and />'p' to be the lines of sight, not 

 appreciably deflected, to the outer edges of the penumbra, 

 P being the edge nearest the centre of the disc, p' the edge 

 nearest the limb. Not taking deflection into account, we 

 see that the penumbral face p u would be absolutely invisible, 

 while the face p'u' would occupy nearly the whole breadth, 

 2Jp', of the range of view. But the lines of sight mu and 

 m'u', being similarly curved by refraction from u and u' to 

 V and u' respectively, the penumbral face pu will mani- 

 festly be brought into view, while the penumbral face p'u' 

 will be naiTowed. And it is easy to undei-stand that since 

 spots vary much in size and depth, and also presumably in 

 the constitution of their vaporous contents, the amount of 

 deflective action will be correspondingly variable, and 

 therefore the varying effects of foreshortening, from that 

 observed by Wilson down to apparent absence of fore- 

 shortening, can be readily explained — or rather are ob- 

 viously to be expected. 



We can see, further, that when the lines of sight to the 

 spot are still more aslant than as shown in fig. -i, the de- 

 flective action maj* just fail to show pu at all, while it will 

 so narrow the opposite face p' u' that this face will also be 

 lost — especially as it is looked at through a much greater 

 range of absorptive vapour. This la.st point must also be 

 considered in dealing with estimates of the two sides of the 

 penumbra of a spot near the edge ; for the darkening of a 

 penumbral streak may very easily be mistaken for narrow- 

 ing ; in fact, from the known laws of irradiation this 

 deceptive eSect may be expected. It would even affect 

 photographs, for photographic irradiation is as real and 

 almost as sensible as the irradiation aflecting the ordinary 

 vision of bright objects. 



SOME PUZZLES. 



{Continued from p. 337.) 



Problem IV. Let aba'b' (fij. I) be an elliptical form, ? the 

 farm lionise. We have to divide this farm into three 

 equal farms, each having a perimeter equal to that of 

 the umUvided farm, and each also in free communica- 

 tiiiii iritli the farm-house. 



^ ^JJUfcJ^^Ij g j^HROTJGH c, the centre of the ellipse, draw 

 the diameter pop'; and divide pcp' into three 

 equal parts in the points p and ji'. On pyi, 

 ! ^^S' i^^ '> ^p' ' describe the half ellipses vhp and pJy', 

 **'''^ v^^^J similar to the half ellipse pabp', and simi- 

 larly situated, On p'// and v'p describe the 

 half ellipses v'b'p' and v'h p, similar to the 

 half ellij.se !■ a'b 1'. and similarly situated. Then the three 

 spaces into which the ellipse is divided by these curved lines 

 are ecjiial to each other, have equal perimeters, and all 

 obviously communicate with the farm p. 



For the half ellipses on the left of pp', taking them in 

 order from p, have areas as 1, 4, 9, so that the portions of 



the farms on that side of pp' have areas as 1,3, 5. The 

 corresponding portions on the other side of pp' have areas 

 as 5, 3, 1, Hence the farms have areas as 1-1-5, 3 -(-3, and 

 5-t-l, or as 6, 6, 6. 



The portions of perimetei's on the left side of pp' are as 

 1, 2, 3 ; the corresponding portions on the right side of pp' 



Fig. 1. 



are as 3, 2, 1. Hence, taking the farms crossed by ?p, pp', 

 and ^p' respectl%'ely, we see (on referring to the figure) that 

 their perimeters are as — 



(i) 1-1-2 + 3, (ii) 2-fl-H-l-2, and(iii) 2 + 1-1-3; 



or as 6, 6, 6 ; while the perimeter of the original farm is as 

 3 + 3, or also as 6. 



It will be obvious that if any straight line be extended 

 across the farms from p, and falling on the left side of pp', 

 it will be divided by the farm boundaries into three equal 

 parts ; and the like with any straight line extended across 

 the farms from p' and falUng on the right side of pp'. 



Problem V. To draw si.c circles cutting each other in eight 

 jwints at an angle of 60'. 



The method is shown in fig. 2. The path to the solution 

 was suggested when the problem was presented as an exer- 

 cise in stereographic projection. 



Fig. 2. 



In fig. 2 the dotted lines represent stereographic projec- 

 tion of the six portions of a sphere corresponding to a 

 circumscribing cube — the six maps, for instance, of the 

 S.D.U.K. star atlas devised by De Morgan — the central 

 point of one of them being the centre of projection. These 

 are shown in dotted outline. The six circles are the stereo- 



