## 2 Method of Angular Correlations  
The first step in using this method is to calculate angular cross correlations on each diffraction pattern in polar coordinates. Polar coordinates are natural for this problem since the particles differ mainly in their orientations (they may also differ in position, and this does not affect the diffraction pattern intensities that are insensitive to the phases of the scattered amplitudes). This is relevant so long as it is a single particle in the beam at one time. Otherwise, the intensities are sensitive for the relative displacements of the particles in the same diffraction pattens. Even in that case, one might hope that, due to the random nature of these displacements, such relative phases are unimportant [8]. Angular pair correlations are related to measured quantities from  
\[C_{2}(q,q^{\prime},\Delta\phi)=<\int I_{p}(q,\phi)I_{p}(q^{\prime},\phi+\Delta \phi)d\Delta\phi>_{p}, \tag{1}\]  
where \(I_{p}(q,\phi)\) is the measured intensity at resolution ring \(q\) and azimuthal angle \(\phi\) on diffraction pattern \(p\), and \(I_{p}(q^{\prime},\phi+\Delta\phi)\) the corresponding intensity at resolution ring \(q^{\prime}\) azimuthal angle \(\phi+\Delta\phi\). Similar to the pair correlations, two-point angular triple correlations may be defined [14]:  
\[C_{3}(q,q^{\prime},\Delta\phi)=<\int I_{p}^{2}(q,\phi)I_{p}(q^{\prime},\phi+ \Delta\phi)d\Delta\phi>_{p}. \tag{2}\]  
The next step is to calculate these quantities to form other quantities related to them in the following way:  
\[B_{I}(q,q^{\prime})=\frac{2l+1}{2}\int C_{2}(q,q^{\prime};\Delta\phi)P_{l}(x)dx \tag{3}\]  
and  
\[T_{I}(q,q^{\prime})=\frac{2l+1}{2}\int C_{3}(q,q^{\prime};\Delta\phi)P_{l}(x)dx, \tag{4}\]  
where \(x=\cos\Delta\phi\). The sequence of operations is summarized next. This method is well suited to the problem of a curved Ewald sphere, as pointed out in, e.g., [11]. A curved Ewald sphere is inevitable at high resolution.The point is that the quantities actually used to construct the 3D diffraction volume, namely, the \(B_{I}(q,q^{\prime})\) and the \(T_{I}(q,q^{\prime})\), are extracted from the quantities measurable in an experiment, namely, the \(C_{2}(q,q^{\prime};\Delta\phi)\) and \(C_{3}(.q,q^{\prime};\Delta\phi)\), by removing the effect of the details of the experiment, such as the X-ray energy, which are contained in the arguments of the Legendre polynomials \(P_{l}\). This allows for a nice separation of the quantities that depend on the experimental details, and the quantities that appear to contain only information about the 3D diffraction volume, which, in turn, gives information about the 3D structure of the particle. These arguments are equally applicable to pair correlation as to the two-point triple correlations defined by [PERSON] [3]. Although [PERSON] [15] and [PERSON] et al. [16] have proposed methods involving a three-point triple correlation function, the two-point triple correlation functions introduced by [PERSON] [14] are easier to measure when photon counts are weak, as pixels with just two non-zero pixels make contributions. If the probability of a high resolution pixel is non-zero is \(10^{-2}\)[12], the probability that two of them are non-zero is \(10^{-4}\). However, reasonable statistics are expected if one has the number of diffraction patterns expected to be measured in a typical experimental shift at even a present-day XFEL. In other words, in [PERSON]'s method [3], one calculated correlations from samples of many randomly oriented molecules. Nevertheless, and this is our crucial point, the correlations are similar to those from a single particle [17] We admit that this demonstration is only for rotations about a single axis, and for delta function like intensities. We investigate here whether it is still likely to hold for a random particle orientation in 3D, and for realistic amplitudes scattered from each particle.