The usual domains for Cauchy distributions have been straight lines and unit circles.These domains are closed under arbitrary changes in location and scale, whether done sequentially or simultaneously.Such closure properties have been extended to spherical Cauchy distributions.Higher dimensional Cauchy based domains are created herein for unit hyperspheres and sets of straight lines of arbitrary dimension, and their Cauchy-like properties are determined and described.Cauchy distributions on these extended domains are shown to be closed under arbitrary transformations of location and scale, done singly or sequentially, but not generally closed when location and scale changes are done simultaneously.Stereographic projections are used to map the curved, finite surface of any hypersphere to a linear, infinite space of the same dimension as that of the hyperspherical surface.These mappings are one-one and onto, with no loss of information.These results show promise for uniting linear and directional mixtures of observations into a common domain linear or directional.

Directional statistics deals with angular data that come from non-linear objects such as circle circumferences or toroidal surfaces.A fundamental problem in directional statistics is that arithmetic cannot be meaningfully done on angles.Naive changes of location and scale like λ' = (λ -µ)/σ for a spherical longitude λ are inappropriate and often misleading since they are not interpretable as one-one mappings from a sphere onto itself.Finding ways to obtain angular scale changes and to construct families of spherical probability distributions that are closed under such scale changes have been unsuccessful.But, such families are successfully constructed herein by indirect but historically powerful methods.Thus, a unit sphere with a uniform probability distribution on its surface is centrally rotated to a suitable position, and then stereographically projected onto an extended complex plane, a linear surface especially amenable to directional and statistical computations.A central dilation is performed on the plane, the dilated plane is projected back in effect as a rescaled sphere, and the rescaled sphere is again rotated.This process induces a family of spherical Cauchy-type probability distributions on the sphere that is closed under composition of such processes (rotate sphere, project sphere to plane, dilate plane, project dilated plane back as a rescaled sphere, and rotate again).The distributions so induced can be generalized to higher dimensional spheres that are also closed under location and scale transformations.These distributions enjoy numerous interrelationships with one another and with linear and circular Cauchy distributions.

Nine girls are presented with variable late systolic murmurs (usually best heard sitting up); five of the nine had mid systolic clicks, minimal to mild mitral regurgitation, and striking variable primary electrocardiographic T wave abnormalities. Data from a tenth girl, who has not yet had cardiac catheterization, with similar clinical features are included. The functional anatomy of the mitral valve and some of the factors that give rise to such a picture are discussed. A detailed cineangiographic analysis of the mitral regurgitation is made with stress on the role of the posterior leaflet and posterior papillary muscle. A quantitative angular analysis of the T waves is made. The vectorcardiogram's superiority to the standard electrogram is made clear in this regard. We believe that this syndrome is caused by posterior papillary muscle dysfunction, though the etiology, including the reason why only girls should be affected, is not known. The prognosis is believed to be good.